Adjectives and commas

WEBVTT
Kind: captions
Language: en

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- [David] Hey grammarians, hey Paige.

00:00:01.768 --> 00:00:02.601
- [Paige] Hi David.

00:00:02.601 --> 00:00:03.434
- [David] Hey, so Paige,

00:00:03.434 --> 00:00:05.155
I went to the grocery store yesterday

00:00:05.155 --> 00:00:08.325
and I got this apple and
I put it in the fridge.

00:00:08.325 --> 00:00:10.105
And this morning, when
I opened the fridge,

00:00:10.105 --> 00:00:14.325
the apple was all gross
and sticky and mushy

00:00:14.325 --> 00:00:16.544
and I really wanna write a
letter to the grocery store

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and say, "Hey, you sold me a gross apple."

00:00:19.305 --> 00:00:20.371
- [Paige] Yeah.

00:00:20.371 --> 00:00:21.574
- [David] But I'm stuck.

00:00:21.574 --> 00:00:22.472
- [Paige] Okay?

00:00:22.472 --> 00:00:26.713
- [David] I feel like I
ought to put a comma in here.

00:00:26.713 --> 00:00:29.894
So, here's the sentence I've got so far.

00:00:29.894 --> 00:00:32.589
"You sold me a mushy, sticky apple!"

00:00:32.589 --> 00:00:33.940
- [Paige] Gross.
- [David] Gross.

00:00:33.940 --> 00:00:35.320
- [David] Unacceptable.
- [Paige] Totally.

00:00:35.320 --> 00:00:37.751
- [David] "So, I would
like my $1.38 back."

00:00:37.751 --> 00:00:39.309
(Paige laughs)

00:00:39.309 --> 00:00:42.298
But in another draft
of the letter, I wrote,

00:00:42.298 --> 00:00:44.525
"You sold me a mushy green apple!"

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- [Paige] Okay, so mushy,
sticky or mushy green.

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- [David] Yeah, and I feel like,

00:00:48.908 --> 00:00:50.810
and I don't know how to
put this into words yet,

00:00:50.810 --> 00:00:52.594
but we're gonna put it into
words 'cause that's our job,

00:00:52.594 --> 00:00:53.577
(Paige laughs)

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whether or not there should be

00:00:54.481 --> 00:00:58.936
a comma between mushy and
green or mushy and sticky,

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'cause these feel different to me.

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- [Paige] Right, right.

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Yeah, so there's actually a couple tests

00:01:05.708 --> 00:01:07.166
that you can do on these sentences

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to see if you need a comma
between the adjectives or not.

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- [David] So, Paige, what distinguishes

00:01:12.945 --> 00:01:16.687
a pair like mushy and
sticky from mushy and green?

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- [Paige] There's this sort of hierarchy

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or order that you put adjectives in

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when you have multiple
of them in a sentence,

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and it is called DOSASCOMP.

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- [David] DOSASCOMP?

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All together now, D

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- [Both] Determiner!

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- [David] O.
- [Both] Opinion!

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- [David] S.
- [Both] Size!

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- [David] A.
- [Both] Age!

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- [David] S again.
- [Both] Shape!

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- [David] C.
- [Both] Color!

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- [David] O.
- [Both] Origin!

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- [David] M.
- [Both] Material!

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- [David] And P.
- [Both] Purpose!

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- [David] Oh, okay.

00:01:47.116 --> 00:01:51.990
So, mushy and sticky are
both opinion adjectives.

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- [Paige] Yeah.

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- [David] So, these are
kinda in the same class,

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whereas mushy and green,
that's an opinion and a color.

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- [Paige] Yeah, they're
in two different classes.

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- [David] Right, so DOSASCOMP helps

00:02:01.929 --> 00:02:04.008
determine adjective order, right?

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- [Paige] Yeah, yeah.

00:02:04.841 --> 00:02:06.273
- [David] So, it's the order
in which you would use,

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if in which you were
stacking these adjectives

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you would use them in
this DOSASCOMP order.

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- [Paige] Right, so like
in the second sentence,

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mushy is an opinion, like you
said, and green is a color.

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And in DOSASCOMP, opinion
comes before color,

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so that's the order
that you write them in.

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So, like in the first sentence,

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when you have mushy and sticky,

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which are both opinion adjectives,

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these are called coordinate adjectives.

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They're in the same category of DOSASCOMP.

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And if you wanted to, you
could reverse the order.

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You could say, "Sticky, mushy apple."

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- [David] Well, let's try that.

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- [Paige] Yeah.

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- [David] Looks good to me.

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Sticky, mushy, mushy, sticky.

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- [Paige] Same thing.

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- [David] Okay, so if
we're trying to determine

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whether or not we have
coordinate adjectives,

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I guess the first thing we would do

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is consult DOSASCOMP, right?

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So that's step one.

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And what is a dosa, Paige?

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- [Paige] Oh, it's like a pancake, right?

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- [David] Yeah, it's like
a South Indian pancake.

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- [Paige] That's pretty cool.

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- [David] And to scomp
is a word we made up

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that means, "to eat."

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- [Paige] Sure, eat some pancakes.

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- [David] Scomp on 'em.

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So, that's step one.

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Step two, try the reversal method.

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- [Paige] Right, and
that's like changing it

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from mushy, sticky to sticky, mushy.

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- [David] And step three
is stick an "and" in there.

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- [Paige] Mm-hmm.

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If you can reverse the
order of the adjectives

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and you can put "and" in
between the two of them,

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then they're coordinate adjectives.

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- [David] And if they're
coordinate adjectives,

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you need to separate them with a comma.

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- [Paige] Exactly.

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And that's why, in the case
of the second sentence,

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with mushy green apple, you
don't put a comma between them.

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They're in different
categories in DOSASCOMP,

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and so they must stay in that order

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and there's no need for a comma.

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- [David] Thanks, Paige.

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I'm gonna get back to
drafting my angry letter.

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(Paige laughs)

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You can learn anything.

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- [David] David out.
- [Paige] Paige out.


Ellipses

WEBVTT
Kind: captions
Language: en

00:00:00.497 --> 00:00:01.375
- [Paige] Hello, grammarians.

00:00:01.375 --> 00:00:02.208
Hello, David.

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- [David] Hello. . . Paige.

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- [Paige] (chuckles) So
in this video we're gonna

00:00:06.692 --> 00:00:10.243
talk about a piece of
punctuation called the ellipsis

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or ellipses in the plural
form as we have here.

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So, what is an ellipsis?

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- [David] So an ellipsis
is a punctuation mark

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that is actually made up of
three periods with spaces

00:00:21.482 --> 00:00:24.315
in between them, boop, boop, boop.

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And it has two main uses.

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- [Paige] Okay.

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- [David] Use number one.

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It indicates a pause in speech.

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So like when you said, "Hello, David."

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And I said, "Hello...Paige."

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- [Paige] Right, so
that ellipsis in the middle

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shows that you're pausing
when you're speaking.

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So like, there's a lot of
reasons why you might pause

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in your speech, you might
be hesitating or thinking

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or just whatever the case may be,

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you can use an ellipsis to
show that you're pausing.

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- [David] Right.

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Uh, and usage number two for the ellipsis

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is that it shows that a portion

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of quoted material has been taken out.

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- [Paige] Why would you
take some of a quote out?

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- [David] So we have here, Paige,

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you have chosen a quote from
the renowned wizard and orator,

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Albus Dumbledore, and he says, "Words are,

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"in my not so humble opinion,

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"our most inexhaustible source of magic."

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If you wanted, you could
kind of take out that aside,

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that, "in my not so humble opinion."

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- [Paige] Yeah, I
don't have a lot of space

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on this screen, so I don't
want to write that whole thing.

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- [David] Mm-hmm.

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- [Paige] Right.

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So I can say, "Words are,
ellipsis, our most inexhaustible

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"source of magic."

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- [David] So, okay, so
Paige, I have a question.

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- [Paige] Yeah.

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- [David] Why do we even need this?

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Why can't you just say,
"Words are our most

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"inexhaustible source of magic"?

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- [Paige] That's a great question.

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So, if I quote someone and
I just sort of willy-nilly

00:01:45.454 --> 00:01:47.973
take words out without
indicating that I've taken

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some information out of the quote,

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I can make it seem like someone said

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something totally different

00:01:53.804 --> 00:01:55.171
to what they actually said, right?

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It can be so easy to misconstrue
someone's actual words.

00:02:01.071 --> 00:02:02.488
So this shows that this isn't exactly

00:02:02.488 --> 00:02:05.746
what Albus Dumbledore
said, but it's the point

00:02:05.746 --> 00:02:08.298
that I want to get across with my quote.

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Does that make sense?

00:02:09.468 --> 00:02:10.301
- [David] I think so.

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So you're saying that there's kind of like

00:02:12.191 --> 00:02:14.848
an ethics question here.

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- [Paige] Right.

00:02:15.681 --> 00:02:18.495
- [David] So how do we
like accurately reproduce

00:02:18.495 --> 00:02:22.705
someone's words without
misrepresenting them?

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- [Paige] Totally.

00:02:23.594 --> 00:02:25.383
- [David] So it shows
that you have messed

00:02:25.383 --> 00:02:26.216
with it somehow.

00:02:26.216 --> 00:02:27.729
- [Paige] Yes, exactly.

00:02:27.729 --> 00:02:29.641
- [David] But that
you're calling out the fact

00:02:29.641 --> 00:02:33.175
that you've cut out pieces, so
that people don't necessarily

00:02:33.175 --> 00:02:34.331
get the wrong idea.

00:02:34.331 --> 00:02:35.556
- [Paige] Totally.

00:02:35.556 --> 00:02:36.901
- [David] But I think
there is kind of an art

00:02:36.901 --> 00:02:38.537
to using ellipsis, right?

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- [Paige] Yeah, I agree.

00:02:39.406 --> 00:02:42.279
You know, you could use it in, you know,

00:02:42.279 --> 00:02:44.723
technically wherever you want
when you're quoting someone

00:02:44.723 --> 00:02:46.263
if you wanna take something out.

00:02:46.263 --> 00:02:50.245
But if you use it over and
over or in the wrong places,

00:02:50.245 --> 00:02:54.219
you can still misrepresent
what a person was saying.

00:02:54.219 --> 00:02:55.079
- [David] If you wanted to,

00:02:55.079 --> 00:02:56.820
you could render that sentence,

00:02:56.820 --> 00:02:58.998
"Words are my opinion."

00:02:58.998 --> 00:02:59.831
- [Paige] Yeah.
- [David] Right.

00:02:59.831 --> 00:03:02.133
- [David] If you didn't use ellipses,

00:03:02.133 --> 00:03:04.399
you could just render
that however you please,

00:03:04.399 --> 00:03:05.966
because those were words that were said

00:03:05.966 --> 00:03:08.547
approximately in that order, right.

00:03:08.547 --> 00:03:10.004
- [Paige] Yeah, without ellipses or--

00:03:10.004 --> 00:03:11.054
- [David] Without ellipses.

00:03:11.054 --> 00:03:11.887
- [Paige] Okay.

00:03:11.887 --> 00:03:12.720
- [David] You know,
you could just be like,

00:03:12.720 --> 00:03:13.999
"Words are my opinion."

00:03:13.999 --> 00:03:14.942
- [Paige] Mm-hmm.

00:03:14.942 --> 00:03:17.034
- [David] And that's
not, that's not exactly

00:03:17.034 --> 00:03:19.008
what Dumbledore said, right?

00:03:19.008 --> 00:03:20.059
- [Paige] Right.

00:03:20.059 --> 00:03:22.055
No, that's not what he was saying at all.

00:03:22.055 --> 00:03:23.142
Those are some of the words he said,

00:03:23.142 --> 00:03:25.521
but that's not the idea he
was trying to get across.

00:03:25.521 --> 00:03:27.881
- [David] So you
have this responsibility

00:03:27.881 --> 00:03:30.955
with ellipses to do the right thing,

00:03:30.955 --> 00:03:35.122
to really represent the way
that somebody speaks accurately.

00:03:36.468 --> 00:03:37.629
- [Paige] Yeah, definitely.

00:03:37.629 --> 00:03:39.607
- [David] Let's talk about the end

00:03:39.607 --> 00:03:42.314
of someone's sentence
and I want to indicate

00:03:42.314 --> 00:03:45.165
that at some point after I cut
off the end of the sentence,

00:03:45.165 --> 00:03:46.665
the sentence ends.

00:03:47.522 --> 00:03:49.439
Where do I put that extra period

00:03:49.439 --> 00:03:50.971
or do I need an extra period?

00:03:50.971 --> 00:03:51.804
- [Paige] Right.

00:03:51.804 --> 00:03:52.780
Okay.

00:03:52.780 --> 00:03:55.901
- [David] So let's say our
sentence is something like,

00:03:55.901 --> 00:03:57.687
"I like cheese, although I'm more partial

00:03:57.687 --> 00:04:00.920
"to a Wensleydale than I am
to a cheddar or a Stilton."

00:04:00.920 --> 00:04:03.328
You might want to just be able
to write that sentence as,

00:04:03.328 --> 00:04:06.578
I like cheese, ellipsis, period, right?

00:04:08.069 --> 00:04:08.902
- [Paige] Yeah.

00:04:08.902 --> 00:04:10.742
- [David] So we're still including

00:04:10.742 --> 00:04:12.942
the terminal punctuation in this sentence.

00:04:12.942 --> 00:04:13.775
- [Paige] Yes.

00:04:13.775 --> 00:04:16.034
- [David] To show that
it ends after the ellipsis.

00:04:16.034 --> 00:04:17.008
- [Paige] Right, yeah.

00:04:17.008 --> 00:04:20.539
That ellipsis there is showing
that stuff has been removed,

00:04:20.539 --> 00:04:24.042
right, from this quote,
and then that final period

00:04:24.042 --> 00:04:25.507
is showing that it's
the end of a sentence,

00:04:25.507 --> 00:04:28.026
just like it normally does.

00:04:28.026 --> 00:04:30.210
'Cause like, the ellipsis, you know,

00:04:30.210 --> 00:04:33.605
looks just like three periods,
but it is it's own thing,

00:04:33.605 --> 00:04:37.128
and doesn't stand in as a period.

00:04:37.128 --> 00:04:38.205
- [David] So that's what,

00:04:38.205 --> 00:04:39.332
so if we didn't have that final period,

00:04:39.332 --> 00:04:41.347
it would just sort of look
like you were trailing off

00:04:41.347 --> 00:04:42.287
like there was a pause.

00:04:42.287 --> 00:04:44.173
I like cheese.

00:04:44.173 --> 00:04:47.443
- [Paige] Yeah, there would
be no end to that sentence.

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- [David] Right.

00:04:48.276 --> 00:04:49.121
But those are the two functions

00:04:49.121 --> 00:04:51.715
of ellipses, then, is that it
indicates a pause in speech,

00:04:51.715 --> 00:04:53.548
like, "Hello...Paige."

00:04:54.473 --> 00:04:56.413
Or it shows that a
portion of quoted material

00:04:56.413 --> 00:04:57.760
has been taken out.

00:04:57.760 --> 00:04:58.593
- [Paige] Yeah.

00:04:58.593 --> 00:05:00.306
- [David] Like, "Words are...our most

00:05:00.306 --> 00:05:02.914
"inexhaustible source of magic."

00:05:02.914 --> 00:05:04.447
- [Paige] That was a
great Dumbledore impression.

00:05:04.447 --> 00:05:06.446
- [Albus] Thank you, Paige.

00:05:06.446 --> 00:05:07.531
You can learn anything.

00:05:07.531 --> 00:05:08.515
- [Paige] Okay.

00:05:08.515 --> 00:05:09.671
- [Albus] Albus out.

00:05:09.671 --> 00:05:11.306
- [Paige] Paige out.
- [David] Cool.


Using quotation marks in titles

WEBVTT
Kind: captions
Language: en

00:00:00.116 --> 00:00:00.956
- [David] Hello, grammarians.

00:00:00.956 --> 00:00:01.789
Hello, Paige.

00:00:01.789 --> 00:00:02.622
- [Paige] Hi, David.

00:00:02.622 --> 00:00:03.455
- [David] So, today we're gonna be

00:00:03.455 --> 00:00:05.065
talking about quotation marks.

00:00:05.065 --> 00:00:06.942
What are they and what do they do?

00:00:06.942 --> 00:00:07.825
Paige Finch.

00:00:07.825 --> 00:00:09.688
- [Paige] We use quotation
marks to indicate

00:00:09.688 --> 00:00:12.360
when someone is speaking, right?

00:00:12.360 --> 00:00:15.724
So, if we're writing dialogue, we can say,

00:00:15.724 --> 00:00:18.937
"I like strawberry jam,"
said Lady Boffington.

00:00:18.937 --> 00:00:20.604
- [David] So, that's one
use of quotation marks,

00:00:20.604 --> 00:00:23.082
which is to quote direct dialogue

00:00:23.082 --> 00:00:26.451
or to quote from a broader work.

00:00:26.451 --> 00:00:30.507
We can also use quotation
marks for the titles of things.

00:00:30.507 --> 00:00:32.767
So, Paige, if you remember,
you can use underlines

00:00:32.767 --> 00:00:36.280
or italics to indicate
the title of something big

00:00:36.280 --> 00:00:39.780
like a book of poetry or an album of songs

00:00:40.712 --> 00:00:43.016
or a movie or a television show.

00:00:43.016 --> 00:00:43.849
- [Paige] Yep.

00:00:43.849 --> 00:00:45.648
- [David] So, Paige, for instance,
one of my favorite albums

00:00:45.648 --> 00:00:49.660
is Gentle Giant's 1975 album Free Hand.

00:00:49.660 --> 00:00:51.717
- [Paige] Okay, but that's
with italics or an underline.

00:00:51.717 --> 00:00:53.837
- [David] It's with italics,
or an underline in this case

00:00:53.837 --> 00:00:55.027
since I'm writing it by hand.

00:00:55.027 --> 00:00:59.194
But track two on that record
is called "On Reflection."

00:01:00.821 --> 00:01:02.564
- [Paige] Okay, so we put quotes

00:01:02.564 --> 00:01:05.166
around each individual song on the album.

00:01:05.166 --> 00:01:06.166
- [David] Right.

00:01:06.166 --> 00:01:07.999
So, this is the album,

00:01:09.273 --> 00:01:12.034
and this is a single song on it.

00:01:12.034 --> 00:01:15.100
Paige, let's say you and Jake
wrote a book of bread poetry.

00:01:15.100 --> 00:01:16.191
- [Paige] Okay, yes.
- [David] Right?

00:01:16.191 --> 00:01:17.264
- [Paige] That is something I would do.

00:01:17.264 --> 00:01:20.330
- [David] And you called
it The Yeast I Can Do.

00:01:20.330 --> 00:01:22.384
- [Paige] That is a great title. (laughs)

00:01:22.384 --> 00:01:24.051
- [David] Thank you.

00:01:25.556 --> 00:01:27.556
- Right, so underline it

00:01:29.768 --> 00:01:31.156
to indicate that that's the full title.

00:01:31.156 --> 00:01:32.906
So, this is the book.

00:01:35.228 --> 00:01:37.230
And then, this book is made
up of individual poems,

00:01:37.230 --> 00:01:39.226
so let's say you wrote a poem in the book

00:01:39.226 --> 00:01:41.559
called "Rye Do You Love Me?"

00:01:48.949 --> 00:01:50.044
Sure, why not?

00:01:50.044 --> 00:01:51.677
So that's in quotes and that indicates

00:01:51.677 --> 00:01:54.109
that this is a single
work or a single poem.

00:01:54.109 --> 00:01:55.327
- [Paige] Okay.

00:01:55.327 --> 00:01:57.090
- [David] Also, let me know
when that book is coming out

00:01:57.090 --> 00:01:58.631
'cause I'll buy your book of poetry.

00:01:58.631 --> 00:01:59.464
(Paige laughs)

00:01:59.464 --> 00:02:02.881
So, it's not just songs and poems, right?

00:02:06.455 --> 00:02:10.288
But it's also magazine
and newspaper articles,

00:02:13.194 --> 00:02:14.194
TV episodes.

00:02:17.748 --> 00:02:21.915
Really, it's anything that is
smaller than a larger work.

00:02:23.394 --> 00:02:26.950
- [Paige] Right, it's something
inside of a larger thing

00:02:26.950 --> 00:02:28.049
like a collection or--

00:02:28.049 --> 00:02:28.882
- [David] Right.

00:02:28.882 --> 00:02:29.716
- [Paige] Yeah, an album.

00:02:29.716 --> 00:02:32.984
- [David] So, if you were
writing for a magazine

00:02:32.984 --> 00:02:36.520
or a newspaper, that newspaper's title,

00:02:36.520 --> 00:02:38.808
the Khan Academy Times,

00:02:38.808 --> 00:02:42.225
would be either italicized or underlined.

00:02:43.634 --> 00:02:47.801
But an article that you wrote
for it would be in quotes.

00:02:49.270 --> 00:02:51.625
I think that about does it
for quotation marks, Paige.

00:02:51.625 --> 00:02:52.474
- [Paige] Yeah, David?

00:02:52.474 --> 00:02:53.307
- [David] Yeah.

00:02:53.307 --> 00:02:55.607
- [Paige] I think I thought
of a bread poetry book name.

00:02:55.607 --> 00:02:56.815
- [David] Okay, what is it?

00:02:56.815 --> 00:02:58.825
- [Paige] Loaves of Grass?

00:02:58.825 --> 00:02:59.778
- [David] Yep.
- [Paige] Yep.

00:02:59.778 --> 00:03:00.814
- [David] Yep, that's pretty good!

00:03:00.814 --> 00:03:02.142
- [Paige] Okay.
- [David] All right.

00:03:02.142 --> 00:03:05.582
Putting it in there.
(Paige laughs)

00:03:05.582 --> 00:03:07.388
That's quotation marks.

00:03:07.388 --> 00:03:09.275
You can learn anything.

00:03:09.275 --> 00:03:10.108
David out.

00:03:10.108 --> 00:03:11.111
- [Paige] Paige out.

00:03:11.111 --> 00:03:12.559
- [David] Loaves of Grass.
(Paige laughs)


Dashes

WEBVTT
Kind: captions
Language: en

00:00:01.161 --> 00:00:02.756
- [Voiceover] Hey,
grammarians, hey, Paige.

00:00:02.756 --> 00:00:03.589
- [Voiceover] Hi, David.

00:00:03.589 --> 00:00:05.170
- [Voiceover] Today we're
gonna talk about dashes

00:00:05.170 --> 00:00:06.993
which is a piece of punctuation,

00:00:06.993 --> 00:00:10.099
looks kind of like this, shoop.

00:00:10.099 --> 00:00:11.792
It's just kind of a straight line.

00:00:11.792 --> 00:00:13.315
Later we're gonna talk about hyphens,

00:00:13.315 --> 00:00:15.221
which look like this, shoop.

00:00:15.221 --> 00:00:17.140
And there is a difference
between the two of them,

00:00:17.140 --> 00:00:17.973
it's kind of confusing,

00:00:17.973 --> 00:00:20.390
but first we're just
gonna talk about dashes,

00:00:20.390 --> 00:00:22.092
what they are, what they do.

00:00:22.092 --> 00:00:23.186
- [Voiceover] Okay, sounds good to me.

00:00:23.186 --> 00:00:26.781
- [Voiceover] So, dashes
have approximately four uses.

00:00:26.781 --> 00:00:29.258
Use number one is that
they can mark interruptions

00:00:29.258 --> 00:00:32.040
in the structure of a sentence, as in,

00:00:32.040 --> 00:00:34.088
"I ran to the bus stop--but the bus

00:00:34.088 --> 00:00:36.023
"had already driven away."

00:00:36.023 --> 00:00:38.959
So we're using this dash
to interrupt the structure

00:00:38.959 --> 00:00:40.774
of this sentence.

00:00:40.774 --> 00:00:44.255
- [Voiceover] Okay, but couldn't
you just put a comma there?

00:00:44.255 --> 00:00:45.615
- [Voiceover] You know,
you're totally could,

00:00:45.615 --> 00:00:48.736
you absolutely could use
a comma instead of a dash.

00:00:48.736 --> 00:00:51.653
But the dash has this feeling.

00:00:51.653 --> 00:00:53.354
And I'm gonna make up a word here,

00:00:53.354 --> 00:00:55.437
it feels more interrupty.

00:00:56.274 --> 00:00:58.226
- [Voiceover] Okay, right,

00:00:58.226 --> 00:01:02.202
so a comma sort of helps
the sentence keep flowing

00:01:02.202 --> 00:01:04.033
and the dash kind of
stops it for a second.

00:01:04.033 --> 00:01:05.578
- [Voiceover] And even
though, grammatically,

00:01:05.578 --> 00:01:07.461
those two sentences are identical, right,

00:01:07.461 --> 00:01:10.077
"I ran to the bus stop, comma,

00:01:10.077 --> 00:01:12.353
"but the bus had already driven away."

00:01:12.353 --> 00:01:13.923
Even though that's, technically,

00:01:13.923 --> 00:01:15.663
that means the exact same thing,

00:01:15.663 --> 00:01:18.369
"As I ran to the bus stop, dash,

00:01:18.369 --> 00:01:20.370
"but the bus had already driven away."

00:01:20.370 --> 00:01:23.462
This is kind of like an abrupt cutoff.

00:01:23.462 --> 00:01:27.172
So, like the comma,
right, the dash is uniting

00:01:27.172 --> 00:01:28.847
these two independent clauses.

00:01:28.847 --> 00:01:31.345
"I ran to the bus stop, the
bus had already driven away."

00:01:31.345 --> 00:01:33.376
Right, so, like, you know, comma,

00:01:33.376 --> 00:01:35.394
but we connect those
two independent clauses

00:01:35.394 --> 00:01:39.496
the same way the dash but is
connecting those two clauses.

00:01:39.496 --> 00:01:41.600
But it has a slightly
different connotation.

00:01:41.600 --> 00:01:43.933
It's just more abrupt in
the way that it connects

00:01:43.933 --> 00:01:45.536
these two things together.

00:01:45.536 --> 00:01:46.369
- [Voiceover] Okay.

00:01:46.369 --> 00:01:49.302
Also, it looks like you don't
have a space before or after

00:01:49.302 --> 00:01:50.746
your dash, is that right?

00:01:50.746 --> 00:01:51.579
- [Voiceover] Right.

00:01:51.579 --> 00:01:53.562
So, originally when I drawn
it, it did look a lot closer.

00:01:53.562 --> 00:01:55.938
But, yes, according to to
the Chicago Manual of Style

00:01:55.938 --> 00:01:58.326
that Khan Academy uses,

00:01:58.326 --> 00:02:02.326
it would just go word,
dash, word, then a space.

00:02:03.330 --> 00:02:06.361
But when you're using dashes, generally,

00:02:06.361 --> 00:02:09.491
you don't put spaces in between them,

00:02:09.491 --> 00:02:10.686
in between words.

00:02:10.686 --> 00:02:12.020
- [Voiceover] Right.

00:02:12.020 --> 00:02:13.233
- [Voiceover] So that's use number one,

00:02:13.233 --> 00:02:15.795
it can interrupt the
structure of a sentence.

00:02:15.795 --> 00:02:18.309
Use number two for the
dash is that they can act

00:02:18.309 --> 00:02:20.476
like parentheses in pairs.

00:02:22.179 --> 00:02:24.283
- [Voiceover] So the
dash is in the same place

00:02:24.283 --> 00:02:27.065
as another type of
punctuation could be again?

00:02:27.065 --> 00:02:27.898
- [Voiceover] Correct.

00:02:27.898 --> 00:02:28.731
- [Voiceover] Okay.

00:02:28.731 --> 00:02:30.925
- [Voiceover] The dash,
I guess, is kind of like,

00:02:30.925 --> 00:02:32.515
so let's look at another example sentence.

00:02:32.515 --> 00:02:33.959
And I'll switch colors.

00:02:33.959 --> 00:02:37.049
"The bug--which landed on
my nose--had to be the size

00:02:37.049 --> 00:02:38.910
"of a softball."

00:02:38.910 --> 00:02:40.082
Enormous bug.

00:02:40.082 --> 00:02:41.811
But, Paige, as you rightly
pointed out, you know,

00:02:41.811 --> 00:02:45.227
this is the same as
having it in parentheses.

00:02:45.227 --> 00:02:47.574
You wouldn't have both of
these together like that.

00:02:47.574 --> 00:02:48.407
- [Voiceover] But they do the same thing.

00:02:48.407 --> 00:02:50.444
- [Voiceover] Yeah, they
have the exact same function.

00:02:50.444 --> 00:02:51.277
- [Voiceover] In this case.

00:02:51.277 --> 00:02:52.110
- [Voiceover] In this case.

00:02:52.110 --> 00:02:53.808
So when you have them
in pairs like so, it's

00:02:53.808 --> 00:02:56.631
"The bug--which landed on
my nose--had to be the size

00:02:56.631 --> 00:02:57.464
"of a softball."

00:02:57.464 --> 00:03:00.089
So it's kind of this aside.

00:03:00.089 --> 00:03:02.437
If we really wanted to,
we could take it right out

00:03:02.437 --> 00:03:04.383
of the sentence, kind
of like an appositive,

00:03:04.383 --> 00:03:05.615
right, with commas.

00:03:05.615 --> 00:03:08.665
So we could say, "The bug had
to be the size of a softball."

00:03:08.665 --> 00:03:09.847
Sentence still makes sense.

00:03:09.847 --> 00:03:10.997
- [Voiceover] Right.

00:03:10.997 --> 00:03:11.907
- [Voiceover] Use number three.

00:03:11.907 --> 00:03:15.927
A dash can be used where you
would otherwise use a colon.

00:03:15.927 --> 00:03:18.165
- [Voiceover] So, in
this case, it's gonna be

00:03:18.165 --> 00:03:19.740
sort of introducing something, right?

00:03:19.740 --> 00:03:20.573
- [Voiceover] Mm-hmm.

00:03:20.573 --> 00:03:22.503
So the colon has this
linking power, right,

00:03:22.503 --> 00:03:24.598
it can introduce ideas,
it can start a list,

00:03:24.598 --> 00:03:27.367
you can do the same thing with a dash.

00:03:27.367 --> 00:03:29.433
So, for example, if you
wanted to introduce an idea,

00:03:29.433 --> 00:03:30.905
we could say.

00:03:30.905 --> 00:03:34.987
- [Voiceover] I have only
ever wanted one thing--snacks.

00:03:34.987 --> 00:03:35.820
- [Voiceover] Me, too, Paige.

00:03:35.820 --> 00:03:36.653
- [Voiceover] So many snacks.

00:03:36.653 --> 00:03:37.988
- [Voiceover] Me, too.

00:03:37.988 --> 00:03:38.821
So, right.

00:03:38.821 --> 00:03:40.973
So we're using this to introduce this idea

00:03:40.973 --> 00:03:44.917
the same way that we
would had it then a colon.

00:03:44.917 --> 00:03:45.750
- [Voiceover] Okay.

00:03:45.750 --> 00:03:47.420
So the dash is like a little actor.

00:03:47.420 --> 00:03:48.671
- [Voiceover] Yeah!

00:03:48.671 --> 00:03:50.001
- [Voiceover] Yeah, the
dash is like an actor

00:03:50.001 --> 00:03:53.133
that can use like do the jobs

00:03:53.133 --> 00:03:56.745
of all these other punctuation marks.

00:03:56.745 --> 00:03:57.578
- [Voiceover] Right.

00:03:57.578 --> 00:03:58.791
- [Voiceover] So it can be parentheses,

00:03:58.791 --> 00:04:00.291
it can be a colon,

00:04:01.424 --> 00:04:04.539
it can be like a comma sometimes.

00:04:04.539 --> 00:04:06.018
- [Voiceover] Wow!

00:04:06.018 --> 00:04:08.085
- [Voiceover] That is
a talented little guy.

00:04:08.085 --> 00:04:09.733
- [Voiceover] Seriously,
I'm very impressed.

00:04:09.733 --> 00:04:12.570
- [Voiceover] And the fourth
use of the dash is, again,

00:04:12.570 --> 00:04:15.199
to indicate interruptions,
but specifically to indicate

00:04:15.199 --> 00:04:17.018
interruptions in speech.

00:04:17.018 --> 00:04:19.388
- [Voiceover] This is like
its own little special thing.

00:04:19.388 --> 00:04:20.221
- [Voiceover] Yeah.

00:04:20.221 --> 00:04:21.054
- [Voiceover] It's not
acting like anything else?

00:04:21.054 --> 00:04:22.298
- [Voiceover] Nope.

00:04:22.298 --> 00:04:25.034
- [Voiceover] Just
indicating an interruption.

00:04:25.034 --> 00:04:26.117
Yes, it does.

00:04:27.141 --> 00:04:27.974
"Watch out for that--"

00:04:27.974 --> 00:04:29.132
- [Voiceover] Wham.

00:04:29.132 --> 00:04:32.132
So we're using this dash to say that

00:04:33.372 --> 00:04:35.315
this sentence was
interrupted by something,

00:04:35.315 --> 00:04:37.358
and then, you know, we
deploy the sound effect.

00:04:37.358 --> 00:04:38.910
But it could be anything, really.

00:04:38.910 --> 00:04:42.547
So if you're trying to write down dialogue

00:04:42.547 --> 00:04:45.118
that is being cutoff by something,

00:04:45.118 --> 00:04:46.455
by another person interrupting,

00:04:46.455 --> 00:04:50.781
by an avalanche of ice
cream, I don't know,

00:04:50.781 --> 00:04:54.497
whatever it is, then you would use a dash

00:04:54.497 --> 00:04:56.064
at the end of the sentence.

00:04:56.064 --> 00:04:57.524
So, no space.

00:04:57.524 --> 00:05:00.857
So we'd go, that, dash, quotation marks.

00:05:01.824 --> 00:05:03.258
And you wouldn't have to do any kind

00:05:03.258 --> 00:05:05.923
of terminal punctuation,
no need for a period

00:05:05.923 --> 00:05:09.506
or a question mark or
an exclamation point.

00:05:11.043 --> 00:05:13.876
The dash kind of serves that role,

00:05:15.350 --> 00:05:17.726
because it's a very fancy actor.

00:05:17.726 --> 00:05:21.268
So I wouldn't say that the
dash as an actor can perform

00:05:21.268 --> 00:05:23.644
the same roles as terminal punctuation,

00:05:23.644 --> 00:05:26.507
as periods or question
marks or exclamation points,

00:05:26.507 --> 00:05:28.512
but rather I would say that
it is cutting off the sentence

00:05:28.512 --> 00:05:30.867
before we can even get to
where those would live.

00:05:30.867 --> 00:05:32.820
- [Voiceover] Okay, yeah, it's saying

00:05:32.820 --> 00:05:34.339
there would've been more
words in this sentence

00:05:34.339 --> 00:05:36.775
and then terminal punctuation, but.

00:05:36.775 --> 00:05:38.736
- [Voiceover] So that's
what dashes can do.

00:05:38.736 --> 00:05:40.640
They are used to mark interruptions

00:05:40.640 --> 00:05:44.279
in the structure of a sentence, like,

00:05:44.279 --> 00:05:46.037
"I ran to the bus stop--but the bus

00:05:46.037 --> 00:05:47.804
"had already driven away."

00:05:47.804 --> 00:05:49.556
- [Voiceover] They can
act like parentheses

00:05:49.556 --> 00:05:51.558
when they're used in pairs, so like,

00:05:51.558 --> 00:05:54.301
"The bug--which landed on
my nose--had to be the size

00:05:54.301 --> 00:05:55.952
"of a softball."

00:05:55.952 --> 00:05:57.520
- [Voiceover] You can
use them like a colon

00:05:57.520 --> 00:06:00.687
to lead into lists or to link an idea.

00:06:01.927 --> 00:06:05.021
So, "I have only ever
wanted one thing--snacks."

00:06:05.021 --> 00:06:07.643
- [Voiceover] And they can
indicate interruptions in speech,

00:06:07.643 --> 00:06:08.775
like, "Watch out for that--"

00:06:08.775 --> 00:06:10.004
- [Voiceover] Wham.

00:06:10.004 --> 00:06:10.925
- [Voiceover] That was good.

00:06:10.925 --> 00:06:12.205
- [Voiceover] Thanks.

00:06:12.205 --> 00:06:14.625
And those are the functions of the dash.

00:06:14.625 --> 00:06:15.809
- [Voiceover] You got it.

00:06:15.809 --> 00:06:17.210
- [Voiceover] You can learn anything.

00:06:17.210 --> 00:06:18.043
David out.

00:06:18.043 --> 00:06:19.062
- [Voiceover] Paige out.


Hyphens vs. dashes

WEBVTT
Kind: captions
Language: en

00:00:00.690 --> 00:00:02.233
- [Voiceover] Hello
grammarians, hello Paige.

00:00:02.233 --> 00:00:03.066
- [Voiceover] Hi, David.

00:00:03.066 --> 00:00:05.182
- [Voiceover] So, today we're
gonna learn about hyphens.

00:00:05.182 --> 00:00:08.631
And what a hyphen is, it's
a little stick, like this,

00:00:08.631 --> 00:00:11.502
as opposed to a dash, which
is about twice as long.

00:00:11.502 --> 00:00:13.537
And people confuse them a lot,

00:00:13.537 --> 00:00:16.271
but they have very different functions.

00:00:16.271 --> 00:00:18.145
So what a hyphen is used to do

00:00:18.145 --> 00:00:22.242
is it's used to join two words into one.

00:00:22.242 --> 00:00:23.275
- [Voiceover] All right.

00:00:23.275 --> 00:00:25.524
- [Voiceover] So, for example,
we have the word yellow.

00:00:25.524 --> 00:00:26.995
That's a word with meaning.

00:00:26.995 --> 00:00:28.082
We have the word green.

00:00:28.082 --> 00:00:29.282
That's a word with meaning.

00:00:29.282 --> 00:00:30.115
- [Voiceover] Yes.

00:00:30.115 --> 00:00:32.484
- [Voiceover] And we may
have a word that does

00:00:32.484 --> 00:00:34.770
fit right in between those two

00:00:34.770 --> 00:00:38.216
or two at once, like
chartreuse or whatever.

00:00:38.216 --> 00:00:39.049
- [Voiceover] Okay.

00:00:39.049 --> 00:00:40.073
- [Voiceover] You know,
but not everybody wants to

00:00:40.073 --> 00:00:41.735
use the word chartreuse.

00:00:41.735 --> 00:00:42.756
- [Voiceover] No, it's kinda crazy.

00:00:42.756 --> 00:00:44.918
- [Voiceover] Kind of a wacky word.

00:00:44.918 --> 00:00:46.018
So, instead of doing that,

00:00:46.018 --> 00:00:48.010
you might just wanna say yellow-green,

00:00:48.010 --> 00:00:51.057
and connect yellow and
green with a hyphen.

00:00:51.057 --> 00:00:55.509
So yeah, so a hyphen is
joining these two things.

00:00:55.509 --> 00:00:58.857
But let's say we had a sentence like,

00:00:58.857 --> 00:01:01.620
"Her hair was yellow --
green were her eyes."

00:01:01.620 --> 00:01:03.364
You know, and so we're still separating

00:01:03.364 --> 00:01:06.487
the words yellow and green with a stick,

00:01:06.487 --> 00:01:08.410
but a dash is longer,

00:01:08.410 --> 00:01:10.785
and instead of uniting yellow-green,

00:01:10.785 --> 00:01:13.793
it's separating them the way

00:01:13.793 --> 00:01:16.377
that you know, a semicolon might.

00:01:16.377 --> 00:01:17.885
- [Voiceover] Yeah, it's not one thing,

00:01:17.885 --> 00:01:20.170
it's she has yellow hair and green eyes.

00:01:20.170 --> 00:01:22.156
- [Voiceover] If she
had yellow-green hair,

00:01:22.156 --> 00:01:23.289
that would be a different story.

00:01:23.289 --> 00:01:24.122
- [Voiceover] Yes.

00:01:24.122 --> 00:01:25.204
- [Voiceover] So what's neat about hyphens

00:01:25.204 --> 00:01:26.844
and what's kinda confusing about hyphens

00:01:26.844 --> 00:01:28.532
is that whether or not we use a hyphen

00:01:28.532 --> 00:01:32.032
indicates how common a compound phrase is.

00:01:33.989 --> 00:01:37.036
So, Brian Gardner in Gardner's
Modern American Usage,

00:01:37.036 --> 00:01:38.747
and in the Chicago Manual of Style

00:01:38.747 --> 00:01:40.984
draws this distinction
with compound words.

00:01:40.984 --> 00:01:44.768
And I should first say what
a compound word is, right?

00:01:44.768 --> 00:01:47.101
A compound word is two words

00:01:48.278 --> 00:01:51.292
smacked together somehow, right?

00:01:51.292 --> 00:01:53.739
So, if we take the...

00:01:53.739 --> 00:01:57.397
so once upon a time, at the
dawn of the Internet page,

00:01:57.397 --> 00:01:58.247
- [Voiceover] Okay.

00:01:58.247 --> 00:02:01.002
- [Voiceover] when
dinosaurs walked the Earth,

00:02:01.002 --> 00:02:02.619
and I was just a wee little baby,

00:02:02.619 --> 00:02:03.494
- [Voiceover] Yes.

00:02:03.494 --> 00:02:07.661
- [Voiceover] you would refer
to getting on line, right.

00:02:09.252 --> 00:02:11.637
And then, as it became
more and more prominent,

00:02:11.637 --> 00:02:12.495
and more and more popular,

00:02:12.495 --> 00:02:15.983
this is what's called an open compound,

00:02:15.983 --> 00:02:19.733
then it became on-line,
separated with a hyphen.

00:02:19.733 --> 00:02:22.744
This is what's called
a hyphenated compound.

00:02:22.744 --> 00:02:23.994
- [Voiceover] Makes sense.

00:02:23.994 --> 00:02:25.548
- [Voiceover] And now,
when we think about it,

00:02:25.548 --> 00:02:27.471
it's just an adjective, and it's online,

00:02:27.471 --> 00:02:29.493
and this is a closed compound.

00:02:29.493 --> 00:02:30.326
- [Voiceover] Right, okay.

00:02:30.326 --> 00:02:34.017
It became closed, cause
everyone knows what that means.

00:02:34.017 --> 00:02:37.089
- [Voiceover] Right, so you
would use hyphenated compounds

00:02:37.089 --> 00:02:39.123
when you're kind of in
this intermediary stage

00:02:39.123 --> 00:02:40.456
of acceptedness.

00:02:41.774 --> 00:02:43.748
- [Voiceover] So like, maybe
one day, in the future,

00:02:43.748 --> 00:02:46.637
yellow-green will be a super common color.

00:02:46.637 --> 00:02:48.984
Right, it's everyone's favorite color,

00:02:48.984 --> 00:02:52.084
so it'll just be smushed
together with no hyphen.

00:02:52.084 --> 00:02:53.098
- [Voiceover] Today is not that day.

00:02:53.098 --> 00:02:54.556
- [Voiceover] Today is not that day,

00:02:54.556 --> 00:02:56.061
so there's a hyphen there.

00:02:56.061 --> 00:02:57.973
- [Voiceover] Right, I
think it's also important

00:02:57.973 --> 00:03:00.348
to hyphenate for clarity.

00:03:00.348 --> 00:03:01.624
Let me explain what I mean by that.

00:03:01.624 --> 00:03:02.774
- [Voiceover] Yeah.

00:03:02.774 --> 00:03:04.239
- [Voiceover] So, first of
all, when I say hyphenate,

00:03:04.239 --> 00:03:06.983
that's just a word that means
to put a hyphen in something.

00:03:06.983 --> 00:03:07.816
- [Voiceover] Yes.

00:03:07.816 --> 00:03:09.257
- [voiceover] To hyphenify.

00:03:09.257 --> 00:03:10.981
So if a compound word could be

00:03:10.981 --> 00:03:13.464
misconstrued or misunderstood,

00:03:13.464 --> 00:03:14.888
you just throw in a hyphen or

00:03:14.888 --> 00:03:16.238
check a dictionary or a style guide.

00:03:16.238 --> 00:03:18.034
So, okay, so, Paige.

00:03:18.034 --> 00:03:18.867
- [Voiceover] Yes.

00:03:18.867 --> 00:03:20.647
- [Voiceover] I would like
to raise some chickens.

00:03:20.647 --> 00:03:21.480
- [Voiceover] Good idea.

00:03:21.480 --> 00:03:23.327
- [Voiceover] So, in order
to raise some chickens,

00:03:23.327 --> 00:03:24.647
I need a chicken coop, right.

00:03:24.647 --> 00:03:26.652
- [Voiceover] That is, yeah, true.

00:03:26.652 --> 00:03:29.125
- [Voiceover] And a coop, right, C-O-O-P,

00:03:29.125 --> 00:03:31.088
is a place where chickens live.

00:03:31.088 --> 00:03:32.129
It's like a little enclosure.

00:03:32.129 --> 00:03:32.962
- [Voiceover] Yeah.

00:03:32.962 --> 00:03:34.368
- [Voiceover] But let's
say I wanna buy this coop

00:03:34.368 --> 00:03:37.499
from a community-run business, right,

00:03:37.499 --> 00:03:42.329
what's called a cooperative
business, or a co-op.

00:03:42.329 --> 00:03:45.563
Well, now sometimes it appears like that.

00:03:45.563 --> 00:03:48.768
Like that, like C-O-O-P, but
if I said something like,

00:03:48.768 --> 00:03:51.562
"I wanna by my coop from
the co-op," like that,

00:03:51.562 --> 00:03:53.033
it'll end up looking like this.

00:03:53.033 --> 00:03:55.031
- [Voiceover] I wanna buy
my coop from the coop.

00:03:55.031 --> 00:03:56.031
- [Voiceover] Right.

00:03:56.031 --> 00:03:58.215
- [Voiceover] Or my
co-op from the co-op, or

00:03:58.215 --> 00:03:59.722
co-op from the coop, or something.

00:03:59.722 --> 00:04:01.894
- [Voiceover] So, in order to be clear,

00:04:01.894 --> 00:04:04.038
it really helps to

00:04:04.038 --> 00:04:06.404
put in that little hyphen.

00:04:06.404 --> 00:04:07.607
And that's the difference, right.

00:04:07.607 --> 00:04:08.775
So you're using this to say,

00:04:08.775 --> 00:04:13.494
oh, you know, this is actually
short for cooperative,

00:04:13.494 --> 00:04:14.631
you know, cause it's operated

00:04:14.631 --> 00:04:16.750
by a bunch of people working together,

00:04:16.750 --> 00:04:17.750
as opposed to this word,

00:04:17.750 --> 00:04:19.749
which is just C-O-O-P.

00:04:19.749 --> 00:04:20.699
- [Voiceover] Right.

00:04:20.699 --> 00:04:21.532
- [voiceover] So you wanna hyphenate

00:04:21.532 --> 00:04:23.786
for clarity when you can.

00:04:23.786 --> 00:04:24.921
- [Voiceover] Makes sense.

00:04:24.921 --> 00:04:27.105
- [Voiceover] So let's bring
all this together, right.

00:04:27.105 --> 00:04:30.978
So I made a yellow-green
coop with the co-op.

00:04:30.978 --> 00:04:31.811
- [Voiceover] Okay,
you all worked together

00:04:31.811 --> 00:04:33.270
to make a yellow-green coop.

00:04:33.270 --> 00:04:34.286
That's pretty cool.

00:04:34.286 --> 00:04:36.504
- [Voiceover] So we're connecting

00:04:36.504 --> 00:04:38.465
yellow and green with a hyphen.

00:04:38.465 --> 00:04:41.025
We're connecting co-op with a hyphen

00:04:41.025 --> 00:04:43.138
to differentiate it from this word coop.

00:04:43.138 --> 00:04:47.163
And that's basically
what you use hyphens for.

00:04:47.163 --> 00:04:49.000
You can learn anything, David out.

00:04:49.000 --> 00:04:50.153
- [Voiceover] Paige out.


Exponential functions differentiation intro

WEBVTT
Kind: captions
Language: en

00:00:00.373 --> 00:00:01.518
- [Voiceover] What I
want to do in this video

00:00:01.518 --> 00:00:05.736
is explore taking the derivatives
of exponential functions.

00:00:05.736 --> 00:00:08.805
So we've already seen that the derivative

00:00:08.805 --> 00:00:12.634
with respect to x of e to the x

00:00:12.634 --> 00:00:14.664
is equal to e to x,

00:00:14.664 --> 00:00:16.557
which is a pretty amazing thing.

00:00:16.557 --> 00:00:19.491
One of the many things that
makes e somewhat special.

00:00:19.491 --> 00:00:21.194
Though when you have an exponential

00:00:21.194 --> 00:00:23.127
with your base right over here as e,

00:00:23.127 --> 00:00:25.494
the derivative of it,
the slope at any point,

00:00:25.494 --> 00:00:29.980
is equal to the value
of that actual function.

00:00:29.980 --> 00:00:33.696
But now let's start exploring
when we have other bases.

00:00:33.696 --> 00:00:37.193
Can we somehow figure out
what is the derivative,

00:00:37.193 --> 00:00:39.739
what is the derivative with respect to x

00:00:39.739 --> 00:00:44.546
when we have a to the x,
where a could be any number?

00:00:44.546 --> 00:00:46.513
Is there some way to figure this out?

00:00:46.513 --> 00:00:50.011
And maybe using our
knowledge that the derivative

00:00:50.011 --> 00:00:52.307
of e to the x, is e to the x.

00:00:52.307 --> 00:00:55.091
Well can we somehow use
a little bit of algebra

00:00:55.091 --> 00:00:57.350
and exponent properties to rewrite this

00:00:57.350 --> 00:01:01.419
so it does look like
something with e as a base?

00:01:01.419 --> 00:01:04.475
Well, you could view a,

00:01:04.475 --> 00:01:08.975
you could view a as being equal to e.

00:01:08.975 --> 00:01:10.133
Let me write it this way.

00:01:10.133 --> 00:01:13.181
Well all right, a as being equal to e

00:01:13.181 --> 00:01:16.493
to the natural log of a.

00:01:16.493 --> 00:01:20.417
Now if this isn't obvious to you,

00:01:20.417 --> 00:01:22.006
I really want you to think about it.

00:01:22.006 --> 00:01:24.350
What is the natural log of a?

00:01:24.350 --> 00:01:28.466
The natural log of a is the power you need

00:01:28.466 --> 00:01:31.973
to raise e to, to get to a.

00:01:31.973 --> 00:01:34.591
So if you actually raise e to that power,

00:01:34.591 --> 00:01:36.660
if you raise e to the power you need

00:01:36.660 --> 00:01:38.636
to raise e too to get to a.

00:01:38.636 --> 00:01:41.069
Well then you're just going to get to a.

00:01:41.069 --> 00:01:43.055
So really think about this.

00:01:43.055 --> 00:01:46.112
Don't just accept this as a leap of faith.

00:01:46.112 --> 00:01:47.476
It should make sense to you.

00:01:47.476 --> 00:01:50.832
And it just comes out of
really what a logarithm is.

00:01:50.832 --> 00:01:55.272
And so we can replace a with
this whole expression here.

00:01:55.272 --> 00:01:59.138
If a is the same thing as
e to the natural log of a,

00:01:59.138 --> 00:02:01.998
well then this is going to be,

00:02:01.998 --> 00:02:06.411
then this is going to be
equal to the derivative

00:02:06.411 --> 00:02:08.449
with respect to x

00:02:08.449 --> 00:02:12.616
of e to the natural log,

00:02:14.065 --> 00:02:16.210
I keep writing la (laughs),

00:02:16.210 --> 00:02:18.368
to the natural log of a

00:02:18.368 --> 00:02:22.427
and then we're going to
raise that to the xth power.

00:02:22.427 --> 00:02:25.075
We're going to raise that to the x power.

00:02:25.075 --> 00:02:27.188
And now this, just using
our exponent properties,

00:02:27.188 --> 00:02:29.884
this is going to be
equal to the derivative

00:02:29.884 --> 00:02:31.884
with respect to x of,

00:02:32.751 --> 00:02:34.691
and I'll just keep color-coding it.

00:02:34.691 --> 00:02:36.307
If I raise something to an exponent

00:02:36.307 --> 00:02:37.605
and then raise that to an exponent,

00:02:37.605 --> 00:02:39.407
that's the same thing as
raising our original base

00:02:39.407 --> 00:02:40.870
to the product of those exponents.

00:02:40.870 --> 00:02:42.745
That's just a basic exponent property.

00:02:42.745 --> 00:02:44.891
So that's going to be the same thing as e

00:02:44.891 --> 00:02:48.435
to the natural log of a,

00:02:48.435 --> 00:02:53.250
natural log of a times x power.

00:02:53.250 --> 00:02:55.833
Times x power.

00:02:56.803 --> 00:02:59.205
And now we can use the chain rule

00:02:59.205 --> 00:03:01.538
to evaluate this derivative.

00:03:02.512 --> 00:03:05.201
So what we will do is

00:03:05.201 --> 00:03:09.368
we will first take the derivative
of the outside function.

00:03:10.772 --> 00:03:14.709
So e to the natural log of a times x

00:03:14.709 --> 00:03:17.108
with respect to the inside function,

00:03:17.108 --> 00:03:21.375
with respect to natural log of a times x.

00:03:21.375 --> 00:03:24.028
And so, this is going to be equal to e

00:03:24.028 --> 00:03:28.769
to the natural log of a times x.

00:03:28.769 --> 00:03:31.692
And then we take the derivative
of that inside function

00:03:31.692 --> 00:03:33.506
with respect to x.

00:03:33.506 --> 00:03:34.655
Well natural log of a,

00:03:34.655 --> 00:03:36.470
it might not immediately jump out to you,

00:03:36.470 --> 00:03:38.264
but that's just going to be a number.

00:03:38.264 --> 00:03:39.855
So that's just going to be,

00:03:39.855 --> 00:03:41.411
so times the derivative.

00:03:41.411 --> 00:03:44.109
If it was the derivative of
three x, it would just be three.

00:03:44.109 --> 00:03:46.491
If it's the derivative
of natural log a times x,

00:03:46.491 --> 00:03:49.957
it's just going to be natural log of a.

00:03:49.957 --> 00:03:52.524
And so this is going to give us

00:03:52.524 --> 00:03:56.922
the natural log of a times e

00:03:56.922 --> 00:03:59.825
to the natural log of a.

00:03:59.825 --> 00:04:01.363
And I'm going to write it like this.

00:04:01.363 --> 00:04:04.640
Natural log of a to the x power.

00:04:04.640 --> 00:04:07.057
Well we've already seen this.

00:04:08.761 --> 00:04:11.393
This right over here is just a.

00:04:11.393 --> 00:04:13.300
So it all simplifies.

00:04:13.300 --> 00:04:17.576
It all simplifies to the natural log of a

00:04:17.576 --> 00:04:21.847
times a to the x,

00:04:21.847 --> 00:04:23.275
which is a pretty neat result.

00:04:23.275 --> 00:04:25.360
So if you're taking the
derivative of e to the x,

00:04:25.360 --> 00:04:27.028
it's just going to be e to the x.

00:04:27.028 --> 00:04:30.984
If you're taking the
derivative of a to the x,

00:04:30.984 --> 00:04:34.668
it's just going to be the natural
log of a times a to the x.

00:04:34.668 --> 00:04:36.644
And so we can now use this result

00:04:36.644 --> 00:04:40.307
to actually take the
derivatives of these types

00:04:40.307 --> 00:04:43.344
of expressions with bases other than e.

00:04:43.344 --> 00:04:46.407
So if I want to find the derivative

00:04:46.407 --> 00:04:48.499
with respect to x

00:04:48.499 --> 00:04:52.860
of eight times three to the x power,

00:04:52.860 --> 00:04:54.504
well what's that going to be?

00:04:54.504 --> 00:04:57.265
Well that's just going to be eight times

00:04:57.265 --> 00:05:00.048
and then the derivative
of this right over here

00:05:00.048 --> 00:05:03.198
is going to be, based on what we just saw,

00:05:03.198 --> 00:05:05.678
it's going to be the
natural log of our base,

00:05:05.678 --> 00:05:09.949
natural log of three times three to the x.

00:05:09.949 --> 00:05:12.004
Times three to the x.

00:05:12.004 --> 00:05:15.099
So it's equal to eight natural log

00:05:15.099 --> 00:05:18.599
of three times three to the x.

00:05:19.534 --> 00:05:23.201
Times three to the x power.


Worked example: Chain rule with table

WEBVTT
Kind: captions
Language: en

00:00:00.403 --> 00:00:02.251
- [Voiceover] The following
table lists the values

00:00:02.251 --> 00:00:03.962
of functions f and g

00:00:03.962 --> 00:00:06.458
and of their derivatives,
f-prime and g-prime

00:00:06.458 --> 00:00:08.945
for the x values negative two and four.

00:00:08.945 --> 00:00:10.751
And so you can see for
x equals negative two,

00:00:10.751 --> 00:00:12.936
x equals four, they gave us the values of

00:00:12.936 --> 00:00:15.500
f, g, f-prime, and g-prime.

00:00:15.500 --> 00:00:19.249
Let function capital-F be
defined as the composition

00:00:19.249 --> 00:00:20.085
of f and g.

00:00:20.085 --> 00:00:22.733
It's lowercase-f of g of x,

00:00:22.733 --> 00:00:26.776
and they want us to
evaluate f-prime of four.

00:00:26.776 --> 00:00:29.309
So you might immediately
recognize that if I have

00:00:29.309 --> 00:00:31.542
a function that can be
viewed as the composition of

00:00:31.542 --> 00:00:35.005
other functions that the
chain rule will apply here.

00:00:35.005 --> 00:00:38.037
And so, and I'm just gonna
restate the chain rule,

00:00:38.037 --> 00:00:39.891
the derivative of capital-F

00:00:39.891 --> 00:00:42.216
is going to be the
derivative of lowercase-f,

00:00:42.216 --> 00:00:45.338
the outside function with
respect to the inside function.

00:00:45.338 --> 00:00:47.838
So lowercase-F-prime of g of x

00:00:49.795 --> 00:00:52.343
times the derivative
of the inside function

00:00:52.343 --> 00:00:56.074
with respect to x times g-prime of x.

00:00:56.074 --> 00:00:58.472
And if we're looking for F-prime of four,

00:00:58.472 --> 00:01:00.566
F-prime of four,

00:01:00.566 --> 00:01:02.648
well everywhere we see an x
we replace it with a four.

00:01:02.648 --> 00:01:06.481
That's gonna be
lowercase-f-prime of g of four

00:01:08.536 --> 00:01:10.369
times g-prime of four.

00:01:12.277 --> 00:01:13.419
Now how do we figure this out?

00:01:13.419 --> 00:01:15.784
They haven't given us
explicitly the values

00:01:15.784 --> 00:01:17.420
of the functions for all xs,

00:01:17.420 --> 00:01:20.427
but they've given it to us
at some interesting points.

00:01:20.427 --> 00:01:22.161
So the first thing you
might wanna figure out is

00:01:22.161 --> 00:01:25.104
well what is g of four going to be?

00:01:25.104 --> 00:01:26.105
Well they tell us:

00:01:26.105 --> 00:01:28.599
when x is equal to four, g of four

00:01:28.599 --> 00:01:30.220
is negative two.

00:01:30.220 --> 00:01:32.701
This tells us that the
value of g of x takes on

00:01:32.701 --> 00:01:35.214
when x is equal to four is negative two.

00:01:35.214 --> 00:01:38.476
So this right over here is negative two.

00:01:38.476 --> 00:01:43.193
And so this first part is
f-prime of negative two.

00:01:43.193 --> 00:01:46.360
So what is f-prime, what is f-prime of

00:01:47.850 --> 00:01:49.276
negative two?

00:01:49.276 --> 00:01:51.845
Well when x is equal to negative two,

00:01:51.845 --> 00:01:53.845
f-prime is equal to one.

00:01:55.330 --> 00:01:59.373
So this right over here is
f-prime of negative two.

00:01:59.373 --> 00:02:01.252
That is equal to one.

00:02:01.252 --> 00:02:04.468
And now we just have to figure
out what g-prime of four is.

00:02:04.468 --> 00:02:06.999
Well when, let me circle this,

00:02:06.999 --> 00:02:08.523
g-prime of four,

00:02:08.523 --> 00:02:12.343
when x is equal to four, and
I'll scroll down a little bit,

00:02:12.343 --> 00:02:16.510
when x is equal to four, g-prime
takes on the value eight.

00:02:17.727 --> 00:02:19.032
So there you have it.

00:02:19.032 --> 00:02:21.819
F-prime of four is
equal to one times eight

00:02:21.819 --> 00:02:25.152
which is equal to eight, and we're done.


Worked example: Derivative of ln(ÃÂx) using the chain rule

WEBVTT
Kind: captions
Language: en

00:00:00.376 --> 00:00:01.591
- [Voiceover] So we have here f of x

00:00:01.591 --> 00:00:06.043
being equal to the natural
log of the square root of x.

00:00:06.043 --> 00:00:07.482
And what we wanna do in this video

00:00:07.482 --> 00:00:09.805
is find the derivative of f.

00:00:09.805 --> 00:00:14.620
And the key here is to
recognize that f can actually

00:00:14.620 --> 00:00:17.526
be viewed as a composition
of two functions.

00:00:17.526 --> 00:00:20.306
And we can diagram that
out, what's going on here?

00:00:20.306 --> 00:00:23.372
Well if you input an
x into our function f,

00:00:23.372 --> 00:00:24.777
what's the first thing that you do?

00:00:24.777 --> 00:00:26.403
Well, you take the square root of it.

00:00:26.403 --> 00:00:30.153
So if we start off with
some x, you input it,

00:00:32.758 --> 00:00:35.890
the first thing that you do,
you take the square root of it.

00:00:35.890 --> 00:00:39.635
You are going to take the
square root of the input

00:00:39.635 --> 00:00:42.302
to produce the square root of x,

00:00:43.181 --> 00:00:44.425
and then what do you do?

00:00:44.425 --> 00:00:46.619
You take the square root and then

00:00:46.619 --> 00:00:48.363
you take the natural log of that.

00:00:48.363 --> 00:00:51.144
So then you take the natural log of that,

00:00:51.144 --> 00:00:53.118
so you could view that as inputting it

00:00:53.118 --> 00:00:55.436
into another function
that takes the natural log

00:00:55.436 --> 00:00:57.769
of whatever is inputted in.

00:00:57.769 --> 00:00:58.847
I'm making these little squares to

00:00:58.847 --> 00:01:00.578
show what you do with the input.

00:01:00.578 --> 00:01:02.062
And then what do you produce?

00:01:02.062 --> 00:01:06.935
Well you produce the natural
log of the square root of x.

00:01:06.935 --> 00:01:10.433
Natural log of the square root of x.

00:01:10.433 --> 00:01:12.358
Which is equal to f of x.

00:01:12.358 --> 00:01:16.275
So you could view f of
x as this entire set,

00:01:18.697 --> 00:01:20.899
or this entire, I guess you could say,

00:01:20.899 --> 00:01:23.963
this combination of
functions right over there.

00:01:23.963 --> 00:01:27.375
That is f of x, which is essentially,

00:01:27.375 --> 00:01:29.756
a composition of two functions.

00:01:29.756 --> 00:01:31.076
You're inputting into one function

00:01:31.076 --> 00:01:34.020
then taking that output and
inputting it into another.

00:01:34.020 --> 00:01:36.237
So you could have a function u here,

00:01:36.237 --> 00:01:38.711
which takes the square root
of whatever its input is,

00:01:38.711 --> 00:01:42.544
so u of x is equal to
the square root of x.

00:01:43.515 --> 00:01:46.015
And then you take that output,

00:01:47.103 --> 00:01:49.972
and input it into another
function that we could call v,

00:01:49.972 --> 00:01:51.365
and what does v do?

00:01:51.365 --> 00:01:53.569
Well it take the natural log
of whatever the input is.

00:01:53.569 --> 00:01:57.259
In this case, in the case
of f, or in the case of how

00:01:57.259 --> 00:02:00.203
I just diagrammed it, v
is taking the natural log,

00:02:00.203 --> 00:02:02.108
the input happens to be square root of x,

00:02:02.108 --> 00:02:05.028
so it outputs the natural
log of the square root of x.

00:02:05.028 --> 00:02:08.198
If we wanted to write
v with x as an input,

00:02:08.198 --> 00:02:10.376
we would just say well
that's the natural log,

00:02:10.376 --> 00:02:13.209
that is just the natural log of x.

00:02:14.070 --> 00:02:16.224
And as you can see here, f of x,

00:02:16.224 --> 00:02:17.979
and I color-coded ahead of time,

00:02:17.979 --> 00:02:20.896
is equal to, f of x is equal to,

00:02:22.006 --> 00:02:24.690
the natural log of the square root of x.

00:02:24.690 --> 00:02:28.857
So that is v of the square
root of x, or v of u of x.

00:02:31.129 --> 00:02:33.971
So it is a composition
which tells you that,

00:02:33.971 --> 00:02:36.024
okay, if I'm trying to
find the derivative here,

00:02:36.024 --> 00:02:39.333
the chain rule is going to be
very, very, very, very useful.

00:02:39.333 --> 00:02:43.074
And the chain rule tells
us that f prime of x

00:02:43.074 --> 00:02:46.574
is going to be equal to the derivative of,

00:02:47.460 --> 00:02:49.061
you can view it as the outside function,

00:02:49.061 --> 00:02:51.421
with respect to this inside function,

00:02:51.421 --> 00:02:54.588
so it's going to be v prime of u of x,

00:02:55.685 --> 00:02:57.185
v prime of u of x,

00:02:58.822 --> 00:03:01.421
times the derivative
of this inside function

00:03:01.421 --> 00:03:02.753
with respect to x.

00:03:02.753 --> 00:03:05.836
So that's just u prime, u prime of x.

00:03:06.677 --> 00:03:08.896
So how do we evaluate these things?

00:03:08.896 --> 00:03:11.292
Well, we know how to take
the derivative of u of x

00:03:11.292 --> 00:03:16.237
and v of x, u prime of x
here, is going to be equal to,

00:03:16.237 --> 00:03:19.691
well remember, square root
of x is just the same thing

00:03:19.691 --> 00:03:22.694
as x to 1/2 power, so we
can use the power rule,

00:03:22.694 --> 00:03:26.688
bring the 1/2 out from so
it becomes 1/2 x to the,

00:03:26.688 --> 00:03:29.420
and then take off one
out of that exponent,

00:03:29.420 --> 00:03:32.606
so that's 1/2 minus one
is negative 1/2 power.

00:03:32.606 --> 00:03:36.606
And what is v of x, sorry,
what is v prime of x?

00:03:38.688 --> 00:03:40.897
Well the derivative of
the natural log of x

00:03:40.897 --> 00:03:45.025
is one over x, we show
that in other videos.

00:03:45.025 --> 00:03:48.325
And so we now know what u prime of x is,

00:03:48.325 --> 00:03:52.942
we know what v prime of x is,
but what is v prime of u of x?

00:03:52.942 --> 00:03:56.775
Well v prime of u of x,
wherever we see the x,

00:03:58.379 --> 00:04:00.879
we replace it, let me write
that a little bit neater,

00:04:00.879 --> 00:04:05.365
we replace that with a u
of x, so v prime of u of x

00:04:05.365 --> 00:04:07.594
is going to be equal to,

00:04:07.594 --> 00:04:10.854
is going to be equal to one over u of x,

00:04:10.854 --> 00:04:13.771
one over u of x, which is equal to,

00:04:15.567 --> 00:04:17.910
which is equal to one over,

00:04:17.910 --> 00:04:20.276
u of x is just the square root of x.

00:04:20.276 --> 00:04:22.776
One over the square root of x.

00:04:24.556 --> 00:04:29.101
This thing right over
here, we have figured out,

00:04:29.101 --> 00:04:31.303
is one over the square root of x,

00:04:31.303 --> 00:04:35.053
and this thing, u prime
of x, we figured out,

00:04:36.393 --> 00:04:39.761
is 1/2 times x to the negative 1/2,

00:04:39.761 --> 00:04:41.780
and x to the negative 1/2,
I could rewrite that as 1/2

00:04:41.780 --> 00:04:46.492
times one over x to the
1/2, which is the same thing

00:04:46.492 --> 00:04:51.159
as 1/2 times one over
the square root of x,

00:04:51.159 --> 00:04:55.241
or I could write that as one
over 2 square roots of x.

00:04:55.241 --> 00:04:58.194
So what is this thing going to be?

00:04:58.194 --> 00:05:02.395
Well this is going to
be equal to, in green,

00:05:02.395 --> 00:05:06.562
v prime of u of x is one
over the square root of x,

00:05:07.501 --> 00:05:11.668
times, times, u prime of
x is one over two times

00:05:13.730 --> 00:05:16.983
the square root of x, now what
is this going to be equal to?

00:05:16.983 --> 00:05:19.794
Well, this is going to be equal to,

00:05:19.794 --> 00:05:21.457
this is just algebra at this point,

00:05:21.457 --> 00:05:24.267
one over, we have our
two and square root of x

00:05:24.267 --> 00:05:26.472
times square root of x is just x.

00:05:26.472 --> 00:05:29.096
So it just simplifies to one over two x.

00:05:29.096 --> 00:05:31.170
So hopefully this made sense,

00:05:31.170 --> 00:05:32.789
and I intentionally diagrammed it out

00:05:32.789 --> 00:05:35.321
so that you start to get that
muscle in your brain going

00:05:35.321 --> 00:05:37.748
of recognizing the composite functions,

00:05:37.748 --> 00:05:39.499
and then making a little bit more sense of

00:05:39.499 --> 00:05:41.201
some of these expressions
of the chain rule

00:05:41.201 --> 00:05:43.333
that you might see in your calculus class,

00:05:43.333 --> 00:05:44.931
or in your calculus textbook.

00:05:44.931 --> 00:05:47.143
But as you get more practice,
you'll be able to do it,

00:05:47.143 --> 00:05:49.994
essentially, without having
to write out all of this.

00:05:49.994 --> 00:05:51.504
You'll say okay, look,
I have a composition.

00:05:51.504 --> 00:05:55.041
This is the natural log
of the square root of x,

00:05:55.041 --> 00:05:56.708
this is v of u of x.

00:05:57.824 --> 00:06:00.050
So what I wanna do is I
wanna take the derivative

00:06:00.050 --> 00:06:02.097
of this outside function with respect

00:06:02.097 --> 00:06:03.942
to this inside function.

00:06:03.942 --> 00:06:07.301
So the derivative of
natural log of something,

00:06:07.301 --> 00:06:11.929
with respect to that something,
is one over that something.

00:06:11.929 --> 00:06:14.287
So it is one over that something,

00:06:14.287 --> 00:06:16.184
the derivative natural log of something

00:06:16.184 --> 00:06:19.624
with respect to that something
is one over that something,

00:06:19.624 --> 00:06:21.182
so that's what we just did here.

00:06:21.182 --> 00:06:24.160
One way to think about it,
what would natural log of x be?

00:06:24.160 --> 00:06:27.571
Well that'd be one over x,
but it's not natural log of x.

00:06:27.571 --> 00:06:28.822
It's one over square root of x,

00:06:28.822 --> 00:06:31.442
so it's going to be one
over the square root of x,

00:06:31.442 --> 00:06:33.540
so you take the derivative
of the outside function

00:06:33.540 --> 00:06:35.158
with respect to the inside one,

00:06:35.158 --> 00:06:37.480
and then you multiply that
times just the derivative

00:06:37.480 --> 00:06:40.897
of the inside function with respect to x.

00:06:41.849 --> 00:06:43.182
And we are done.


Worked example: Derivative of cos_(x) using the chain rule

WEBVTT
Kind: captions
Language: en

00:00:00.373 --> 00:00:02.927
- [Voiceover] Let's say we
have the function f of x

00:00:02.927 --> 00:00:07.094
which is equal to cosine
of x to the third power

00:00:08.477 --> 00:00:10.561
which we could also write like this,

00:00:10.561 --> 00:00:13.394
cosine of x to the third power.

00:00:14.846 --> 00:00:16.892
And we are interested in figuring out

00:00:16.892 --> 00:00:21.139
what f prime of x is going to be equal to.

00:00:21.139 --> 00:00:25.751
So we want to figure out f
prime of x and as we will see,

00:00:25.751 --> 00:00:28.574
the chain rule is going
to be very useful here

00:00:28.574 --> 00:00:29.407
and what I'm going to do is

00:00:29.407 --> 00:00:31.235
I'm going to first just
apply the chain rule

00:00:31.235 --> 00:00:32.489
and then maybe dig into it a little bit

00:00:32.489 --> 00:00:34.052
to make sure we draw the connection

00:00:34.052 --> 00:00:36.327
between what we're doing here
and then what you might see

00:00:36.327 --> 00:00:38.163
in maybe some of your Calculus textbooks

00:00:38.163 --> 00:00:39.958
that explain the chain rule.

00:00:39.958 --> 00:00:42.328
So if we have a function

00:00:42.328 --> 00:00:45.264
that is defined as essentially
a composite function,

00:00:45.264 --> 00:00:47.880
notice this expression right here,

00:00:47.880 --> 00:00:50.410
we are taking something
to the third power.

00:00:50.410 --> 00:00:52.534
It isn't just an x that we're
taking to the third power.

00:00:52.534 --> 00:00:55.475
We are taking a cosine
of x to the third power.

00:00:55.475 --> 00:00:57.601
So we're taking a function,
you could view it this way,

00:00:57.601 --> 00:00:59.643
we're taking the function cosine of x

00:00:59.643 --> 00:01:01.847
and then we're inputting
it in to another function

00:01:01.847 --> 00:01:04.788
that takes it to the third power.

00:01:04.788 --> 00:01:06.218
So let me put it this way.

00:01:06.218 --> 00:01:07.940
If you viewed,

00:01:07.940 --> 00:01:10.430
if you say, look, we could take an x,

00:01:10.430 --> 00:01:14.430
we put it into one function and that is,

00:01:14.430 --> 00:01:17.086
that first function is cosine of x

00:01:17.086 --> 00:01:19.753
so first, we evaluate the cosine

00:01:21.008 --> 00:01:24.520
and so that's going to
produce cosine of x,

00:01:24.520 --> 00:01:26.316
cosine of x,

00:01:26.316 --> 00:01:28.804
and then we're going to
input it into a function

00:01:28.804 --> 00:01:31.301
that just takes things to the third power.

00:01:31.301 --> 00:01:34.365
So it just takes things
to the third power.

00:01:34.365 --> 00:01:36.806
And so what are you going to end up with?

00:01:36.806 --> 00:01:38.263
Well, you're going to end up with,

00:01:38.263 --> 00:01:39.572
what are you taking to the third power?

00:01:39.572 --> 00:01:41.741
You're taking cosine of x.

00:01:41.741 --> 00:01:44.840
Cosine of x to the third power.

00:01:44.840 --> 00:01:47.763
This is a composite function.

00:01:47.763 --> 00:01:50.022
You could view this,

00:01:50.022 --> 00:01:52.945
you could view this as the function,

00:01:52.945 --> 00:01:55.476
let's call this blue one, the function v

00:01:55.476 --> 00:01:58.743
and let's call this the function u

00:01:58.743 --> 00:02:01.679
and so if we're taking x and into u,

00:02:01.679 --> 00:02:02.846
this is u of x

00:02:04.251 --> 00:02:07.273
and then if we're taking
u of x into the input

00:02:07.273 --> 00:02:09.600
or as the input into the function v

00:02:09.600 --> 00:02:11.787
then this output right over here,

00:02:11.787 --> 00:02:13.870
this is going to be v of,

00:02:15.562 --> 00:02:16.910
well, what was inputted?

00:02:16.910 --> 00:02:17.910
V of u of x.

00:02:18.872 --> 00:02:19.789
V of u of x

00:02:21.932 --> 00:02:23.528
or another way of writing it,

00:02:23.528 --> 00:02:24.713
I'm going to write it multiple ways.

00:02:24.713 --> 00:02:28.430
That's the same thing as v of cosine of x.

00:02:28.430 --> 00:02:29.847
V of cosine of x.

00:02:31.535 --> 00:02:33.619
And so v, whatever you input into it,

00:02:33.619 --> 00:02:35.336
it just takes it to the third power.

00:02:35.336 --> 00:02:36.358
If you were to write v of x,

00:02:36.358 --> 00:02:38.634
it would be x to the third power.

00:02:38.634 --> 00:02:40.036
So the chain rule tells us

00:02:40.036 --> 00:02:41.997
or the chain rule is what
our brain should say.

00:02:41.997 --> 00:02:44.203
Hey, it becomes applicable

00:02:44.203 --> 00:02:46.203
if we're going to take the
derivative of a function

00:02:46.203 --> 00:02:49.753
that can be expressed as a
composite function like this.

00:02:49.753 --> 00:02:53.795
So just to be clear, we can write f of x.

00:02:53.795 --> 00:02:56.628
f of x is equal to v of u of x.

00:02:57.919 --> 00:03:00.232
I know I'm essentially
saying the same thing

00:03:00.232 --> 00:03:01.148
over and over again

00:03:01.148 --> 00:03:02.324
but I'm saying it in
slightly different ways

00:03:02.324 --> 00:03:03.755
because the first time you learn this,

00:03:03.755 --> 00:03:05.794
it can be a little bit hard to grok

00:03:05.794 --> 00:03:08.042
or really deeply understand

00:03:08.042 --> 00:03:10.738
so I'm going to try to
write it in different ways.

00:03:10.738 --> 00:03:12.699
And the chain rule tells us

00:03:12.699 --> 00:03:14.862
that if you have a situation like this

00:03:14.862 --> 00:03:17.960
then the derivative, f prime of x,

00:03:17.960 --> 00:03:20.247
and this is something that you
will see in your textbooks.

00:03:20.247 --> 00:03:21.541
Well, this is going to be

00:03:21.541 --> 00:03:24.490
the derivative of this whole thing

00:03:24.490 --> 00:03:25.920
with respect to u of x

00:03:25.920 --> 00:03:29.587
so we could write that
as v prime of u of x.

00:03:30.614 --> 00:03:32.031
V prime of u of x

00:03:33.749 --> 00:03:37.247
times the derivative
of u with respect to x.

00:03:37.247 --> 00:03:38.830
Times u prime of x.

00:03:40.307 --> 00:03:41.411
This right over here,

00:03:41.411 --> 00:03:45.379
this is one expression of the chain rule

00:03:45.379 --> 00:03:48.637
and so how do we evaluate it in this case?

00:03:48.637 --> 00:03:50.927
Well, let me color code
it in a similar way.

00:03:50.927 --> 00:03:52.889
So the v function,

00:03:52.889 --> 00:03:54.932
this outer thing that just
takes things to the third power,

00:03:54.932 --> 00:03:56.034
I'll put in blue.

00:03:56.034 --> 00:03:57.709
So f prime of x,

00:03:57.709 --> 00:03:59.788
another way of expressing it

00:03:59.788 --> 00:04:02.360
and I'll use it with more of
the differential notation,

00:04:02.360 --> 00:04:06.367
you could view this as the derivative of,

00:04:06.367 --> 00:04:07.955
well, I'll write it a
couple of different ways.

00:04:07.955 --> 00:04:11.965
You could view it as the derivative of v.

00:04:11.965 --> 00:04:13.548
The derivative of v

00:04:16.095 --> 00:04:17.595
with respect to u.

00:04:19.189 --> 00:04:20.618
I want to get the colors right.

00:04:20.618 --> 00:04:23.785
The derivative of v with respect to u,

00:04:26.361 --> 00:04:28.940
that's what this thing is right over here,

00:04:28.940 --> 00:04:31.023
times the derivative of u

00:04:32.178 --> 00:04:33.678
with respect to x.

00:04:35.250 --> 00:04:39.114
So times the derivative
of u with respect to x.

00:04:39.114 --> 00:04:40.185
And just to be clear,

00:04:40.185 --> 00:04:41.774
so you're familiar with
the different notations

00:04:41.774 --> 00:04:43.388
you'll see in different textbooks,

00:04:43.388 --> 00:04:47.943
this is this right over here
just using different notations

00:04:47.943 --> 00:04:50.693
and this is this right over here.

00:04:51.861 --> 00:04:53.777
So let's actually evaluate these things.

00:04:53.777 --> 00:04:56.730
You're probably tired of
just talking in the abstract.

00:04:56.730 --> 00:04:59.649
So this is going to be equal to,

00:04:59.649 --> 00:05:01.533
this is going to be equal to

00:05:01.533 --> 00:05:02.762
and I'm going to write it out again,

00:05:02.762 --> 00:05:05.007
this is the derivative,

00:05:05.007 --> 00:05:06.724
instead of just writing v and u,

00:05:06.724 --> 00:05:08.928
I'm going to write it,
let me write this way.

00:05:08.928 --> 00:05:10.697
This is going to be,

00:05:10.697 --> 00:05:13.056
I keep wanting, I'm
using the wrong colors.

00:05:13.056 --> 00:05:15.790
This is going to be the derivative of,

00:05:15.790 --> 00:05:17.057
and I'm going to leave some space,

00:05:17.057 --> 00:05:20.070
times the derivative of something else

00:05:20.070 --> 00:05:22.121
with respect to something else

00:05:22.121 --> 00:05:23.820
so we're going to have to
first take the derivative of v.

00:05:23.820 --> 00:05:24.653
Well, v is

00:05:26.250 --> 00:05:29.191
cosine of x to the third power.

00:05:29.191 --> 00:05:30.344
Cosine of x.

00:05:30.344 --> 00:05:31.474
We're going to take the derivative of that

00:05:31.474 --> 00:05:35.287
with respect to u which
is just cosine of x

00:05:35.287 --> 00:05:36.267
and we're going to multiply that

00:05:36.267 --> 00:05:40.140
times the derivative of
u which is cosine of x

00:05:40.140 --> 00:05:42.230
with respect to x.

00:05:42.230 --> 00:05:44.519
With respect to x.

00:05:44.519 --> 00:05:46.686
So this one, we have good,

00:05:47.943 --> 00:05:50.117
we've seen this before.

00:05:50.117 --> 00:05:53.129
We know that the derivative
with respect to x

00:05:53.129 --> 00:05:54.379
of cosine of x.

00:05:56.284 --> 00:05:57.525
Cosine.

00:05:57.525 --> 00:05:59.728
We use it in that same color.

00:05:59.728 --> 00:06:02.666
The derivative of cosine of x,

00:06:02.666 --> 00:06:05.529
well, that's equal to negative sine of x.

00:06:05.529 --> 00:06:09.693
So this one right over here,
that is negative sine of x.

00:06:09.693 --> 00:06:11.247
You might be more familiar with seeing

00:06:11.247 --> 00:06:13.004
the derivative operated this way

00:06:13.004 --> 00:06:15.536
but in theory, you won't see this as often

00:06:15.536 --> 00:06:17.906
but this helps my brain
really grok what we're doing.

00:06:17.906 --> 00:06:19.295
We're taking the derivative of cosine of x

00:06:19.295 --> 00:06:20.845
with respect to x.

00:06:20.845 --> 00:06:22.681
Well, that's going to
be negative sine of x.

00:06:22.681 --> 00:06:24.008
Well, what about taking the derivative

00:06:24.008 --> 00:06:26.831
of cosine of x to the third power

00:06:26.831 --> 00:06:29.005
with respect to cosine of x?

00:06:29.005 --> 00:06:31.647
What is this thing over here mean?

00:06:31.647 --> 00:06:34.067
Well, if I were taking the derivative,

00:06:34.067 --> 00:06:37.701
if I was taking the derivative of,

00:06:37.701 --> 00:06:38.534
let me write it this way,

00:06:38.534 --> 00:06:43.122
if I was taking the derivative
of x to the third power,

00:06:43.122 --> 00:06:46.839
x to the third power with respect to x,

00:06:46.839 --> 00:06:48.296
if it was like that,

00:06:48.296 --> 00:06:50.435
well, this is just going to be

00:06:50.435 --> 00:06:51.538
and let me put some brackets here

00:06:51.538 --> 00:06:52.768
to make it a little bit clear.

00:06:52.768 --> 00:06:55.546
If I'm taking the derivative of that,

00:06:55.546 --> 00:06:57.327
that is going to be,

00:06:57.327 --> 00:06:58.160
that is going to be,

00:06:58.160 --> 00:06:59.711
we bring the exponent out front.

00:06:59.711 --> 00:07:01.705
That's going to be three,

00:07:01.705 --> 00:07:02.872
three times x.

00:07:05.185 --> 00:07:08.446
Three times x to the second power.

00:07:08.446 --> 00:07:12.082
Three times x to the second power.

00:07:12.082 --> 00:07:14.386
So the general notion here is

00:07:14.386 --> 00:07:17.426
if I'm taking the derivative of something,

00:07:17.426 --> 00:07:19.060
whatever this something happens to be,

00:07:19.060 --> 00:07:20.287
let me do this in a new color.

00:07:20.287 --> 00:07:24.371
I'm doing the derivative of
orange circle to the third power

00:07:24.371 --> 00:07:27.159
with respect to orange circle.

00:07:27.159 --> 00:07:29.804
Well, that's just going
to be three times orange

00:07:29.804 --> 00:07:31.195
or yellow circle.

00:07:31.195 --> 00:07:33.116
Let me make it an actual orange circle.

00:07:33.116 --> 00:07:36.225
So the derivative of orange
circle to the third power

00:07:36.225 --> 00:07:38.834
with respect to orange circle,

00:07:38.834 --> 00:07:42.344
that's going to be three times
the orange circle squared.

00:07:42.344 --> 00:07:44.101
So if I'm taking the
derivative of cosine of x

00:07:44.101 --> 00:07:47.293
to the third power with
respect to cosine of x,

00:07:47.293 --> 00:07:48.928
well, that's just going to be,

00:07:48.928 --> 00:07:50.928
this is just going to be

00:07:51.797 --> 00:07:53.797
three times cosine of x,

00:07:55.407 --> 00:07:58.324
cosine of x to the second power.

00:07:59.452 --> 00:08:01.119
To the second power.

00:08:02.642 --> 00:08:04.072
Notice, one way to think about it.

00:08:04.072 --> 00:08:06.399
I'm taking the derivative
of this outside function

00:08:06.399 --> 00:08:07.991
with respect to the inside.

00:08:07.991 --> 00:08:09.623
So I would do the same thing

00:08:09.623 --> 00:08:11.501
as taking the derivative
of x to the third power

00:08:11.501 --> 00:08:13.299
but instead of an x, I have a cosine of x

00:08:13.299 --> 00:08:14.969
so instead of it being three x squared,

00:08:14.969 --> 00:08:18.281
it is three cosine of x squared

00:08:18.281 --> 00:08:19.540
and then the chain rule says,

00:08:19.540 --> 00:08:20.690
if we want to get finally get

00:08:20.690 --> 00:08:21.792
the derivative with respect to x,

00:08:21.792 --> 00:08:24.037
we then take the derivative of cosine of x

00:08:24.037 --> 00:08:25.800
with respect to x.

00:08:25.800 --> 00:08:29.223
I know that's a big mouthful
but we are at the homestretch.

00:08:29.223 --> 00:08:31.593
We've now figured out the derivative.

00:08:31.593 --> 00:08:34.010
It's going to be this times this.

00:08:34.010 --> 00:08:36.781
So let's see, that's going
to be negative three,

00:08:36.781 --> 00:08:39.281
negative three times sine of x

00:08:40.711 --> 00:08:42.794
times cosine squared of x

00:08:45.735 --> 00:08:47.938
and I know that was kind
of a long way of saying it.

00:08:47.938 --> 00:08:50.348
I'm trying to explain the
chain rule at the same time.

00:08:50.348 --> 00:08:52.143
But once you get the hang of
it, you're just going to say,

00:08:52.143 --> 00:08:56.170
alright, well, let me take
the derivative of the outside

00:08:56.170 --> 00:08:59.472
of something to the third power
with respect to the inside.

00:08:59.472 --> 00:09:02.488
Let me just treat that cosine
of x like as if it was an x.

00:09:02.488 --> 00:09:03.349
Well, that's going to be,

00:09:03.349 --> 00:09:06.819
if I do that, that's going to
be three cosine squared of x

00:09:06.819 --> 00:09:08.815
so that's that part and that part

00:09:08.815 --> 00:09:11.467
and then let me take the
derivative of the inside

00:09:11.467 --> 00:09:12.539
with respect to x.

00:09:12.539 --> 00:09:15.289
Well, that is negative sine of x.


Derivatives of sec(x) and csc(x)

WEBVTT
Kind: captions
Language: en

00:00:00.247 --> 00:00:02.981
- [Voiceover] In a previous
video we used the quotient rule

00:00:02.981 --> 00:00:04.489
in order to find the derivatives

00:00:04.489 --> 00:00:06.687
of tangent of x and cotangnet of x.

00:00:06.687 --> 00:00:09.221
And what I what to do in
this video is to keep going

00:00:09.221 --> 00:00:13.479
and find the derivatives of
secant of x and cosecant of x.

00:00:13.479 --> 00:00:15.140
So let's start with secant of x.

00:00:15.140 --> 00:00:19.735
The derivative with respect
to x of secant of x.

00:00:19.735 --> 00:00:22.279
Well, secant of x is the same thing as

00:00:22.279 --> 00:00:24.814
so we're going to find the
derivative with respect to x

00:00:24.814 --> 00:00:27.695
of secant of x is the same thing as

00:00:27.695 --> 00:00:30.077
one over,

00:00:30.077 --> 00:00:32.859
one over the cosine of x.

00:00:32.859 --> 00:00:35.526
And that's just the definition of secant.

00:00:35.526 --> 00:00:38.000
And there's multiple
ways you could do this.

00:00:38.000 --> 00:00:39.541
When you learn the chain rule,

00:00:39.541 --> 00:00:41.908
that actually might be a
more natural thing to use

00:00:41.908 --> 00:00:43.515
to evaluate the derivative here.

00:00:43.515 --> 00:00:45.284
But we know the quotient rule,

00:00:45.284 --> 00:00:47.158
so we will apply the quotient rule here.

00:00:47.158 --> 00:00:49.035
And it's no coincidence that
you get to the same answer.

00:00:49.035 --> 00:00:51.743
The quotient rule actually can be derived

00:00:51.743 --> 00:00:54.418
based on the chain rule
and the product rule.

00:00:54.418 --> 00:00:55.749
But I won't keep going into that.

00:00:55.749 --> 00:00:58.096
Let's just apply the quotient
rule right over here.

00:00:58.096 --> 00:01:01.709
So this derivative is
going to be equal to,

00:01:01.709 --> 00:01:05.052
it's going to be equal to
the derivative of the top.

00:01:05.052 --> 00:01:07.462
Well, what's the derivative
of one with respect to x?

00:01:07.462 --> 00:01:09.363
Well, that's just zero.

00:01:09.363 --> 00:01:11.917
Times the function on the bottom.

00:01:11.917 --> 00:01:15.361
So, times cosine of x.

00:01:15.361 --> 00:01:16.993
Cosine of x.

00:01:16.993 --> 00:01:18.642
Minus,

00:01:18.642 --> 00:01:22.768
minus the function on the top.

00:01:22.768 --> 00:01:24.940
Well, that's just one.

00:01:24.940 --> 00:01:27.368
Times the derivative on the bottom.

00:01:27.368 --> 00:01:29.444
Well, the derivative on the bottom is,

00:01:29.444 --> 00:01:31.863
the derivative of cosine
of x is negative sine of x.

00:01:31.863 --> 00:01:33.967
So we could put the sine of x there.

00:01:33.967 --> 00:01:35.027
But it's negative sine of x,

00:01:35.027 --> 00:01:37.691
so you have a minus and
it'll be a negative,

00:01:37.691 --> 00:01:39.891
so we can just make that a positive.

00:01:39.891 --> 00:01:42.492
And then all of that over the function

00:01:42.492 --> 00:01:44.040
on the bottom squared.

00:01:44.040 --> 00:01:47.429
So, cosine of x, squared.

00:01:47.429 --> 00:01:49.107
And so zero times cosine of x,

00:01:49.107 --> 00:01:50.916
that is just zero.

00:01:50.916 --> 00:01:55.582
And so all we are left with is sine of x

00:01:55.582 --> 00:01:59.057
over cosine of x squared.

00:01:59.057 --> 00:02:01.440
And there's multiple ways
that you could rewrite this

00:02:01.440 --> 00:02:02.501
if you like.

00:02:02.501 --> 00:02:07.209
You could say that this
is same thing as sine of x

00:02:07.209 --> 00:02:11.376
over cosine of x times
one over cosine of x.

00:02:13.201 --> 00:02:16.118
And of course this is tangent of x,

00:02:17.028 --> 00:02:19.698
times secant of x.

00:02:19.698 --> 00:02:20.719
Secant of x.

00:02:20.719 --> 00:02:22.422
So you could say derivative
of secant of x is

00:02:22.422 --> 00:02:24.693
sine of x over cosine-squared of x.

00:02:24.693 --> 00:02:28.610
Or it is tangent of x
times the secant of x.

00:02:29.650 --> 00:02:31.564
So now let's do cosecant.

00:02:31.564 --> 00:02:35.731
So the derivative with
respect to x of cosecant of x.

00:02:37.268 --> 00:02:39.179
Well, that's the same
thing as the derivative

00:02:39.179 --> 00:02:43.346
with respect to x of one over sine of x.

00:02:45.174 --> 00:02:48.213
Cosecant is one over sine of x.

00:02:48.213 --> 00:02:50.540
I remember that because
you think it's cosecant.

00:02:50.540 --> 00:02:54.672
Maybe it's the reciprocal
of cosine, but it's not.

00:02:54.672 --> 00:02:56.705
It's the opposite of
what you would expect.

00:02:56.705 --> 00:02:59.717
Cosine's reciprocal isn't
cosecant, it is secant.

00:02:59.717 --> 00:03:02.805
Once again, opposite of
what you would expect.

00:03:02.805 --> 00:03:04.543
That starts with an s,
this starts with a c.

00:03:04.543 --> 00:03:06.603
That starts with a c,
that starts with an s.

00:03:06.603 --> 00:03:08.546
It's just way it happened to be defined.

00:03:08.546 --> 00:03:10.294
But anyway, let's just evaluate this.

00:03:10.294 --> 00:03:11.542
Once again, we'll do the quotient rule,

00:03:11.542 --> 00:03:13.811
but you could also do
this using the chain rule.

00:03:13.811 --> 00:03:15.103
So it's going to be

00:03:15.103 --> 00:03:19.487
the derivative of the expression
on top, which is zero,

00:03:19.487 --> 00:03:23.654
times the expression on the
bottom, which is sine of x.

00:03:24.764 --> 00:03:26.504
Sine of x.

00:03:26.504 --> 00:03:30.671
Minus the expression on
top, which is just one.

00:03:31.640 --> 00:03:34.424
Times the derivative of the
expression on the bottom,

00:03:34.424 --> 00:03:36.174
which is cosine of x.

00:03:37.056 --> 00:03:40.349
All of that over the expression
on the bottom squared.

00:03:40.349 --> 00:03:42.752
Sine-squared of x.

00:03:42.752 --> 00:03:44.106
That's zero.

00:03:44.106 --> 00:03:46.863
So we get negative cosine of x

00:03:46.863 --> 00:03:49.780
over sine-squared of x.

00:03:51.641 --> 00:03:54.134
So that's one way to think about it.

00:03:54.134 --> 00:03:56.745
Or if you like, you could do this,

00:03:56.745 --> 00:03:58.157
the same thing we did over here,

00:03:58.157 --> 00:04:00.869
this is the same thing
as negative cosine of x

00:04:00.869 --> 00:04:05.036
over sine of x, times one over sine of x.

00:04:06.019 --> 00:04:10.517
And this is negative cotangent of x.

00:04:10.517 --> 00:04:13.100
Negative cotangent of x, times,

00:04:14.392 --> 00:04:16.020
maybe I'll write it this way,

00:04:16.020 --> 00:04:19.545
times one over sine of x is cosecant of x.

00:04:19.545 --> 00:04:22.457
Cosecant of x.

00:04:22.457 --> 00:04:25.790
So, which ever one you find more useful.


Derivatives of tan(x) and cot(x)

WEBVTT
Kind: captions
Language: en

00:00:00.164 --> 00:00:01.504
- [Voiceover] We already
know the derivatives

00:00:01.504 --> 00:00:02.723
of sine and cosine.

00:00:02.723 --> 00:00:05.295
We know that the derivative
with respect to x

00:00:05.295 --> 00:00:08.628
of sine of x is equal to cosine of x.

00:00:10.808 --> 00:00:14.392
We know that the derivative
with respect to x

00:00:14.392 --> 00:00:18.475
of cosine of x is equal
to negative sine of x.

00:00:20.237 --> 00:00:21.997
And so what we want to do in this video

00:00:21.997 --> 00:00:26.078
is find the derivatives of the
other basic trig functions.

00:00:26.078 --> 00:00:28.646
So, in particular, we
know, let's figure out

00:00:28.646 --> 00:00:31.322
what the derivative with respect to x,

00:00:31.322 --> 00:00:33.677
let's first do tangent of x.

00:00:33.677 --> 00:00:36.817
Tangent of x, well this is the same thing

00:00:36.817 --> 00:00:40.984
as trying to find the
derivative with respect to x of,

00:00:42.489 --> 00:00:46.247
well, tangent of x is just sine of x,

00:00:46.247 --> 00:00:48.747
sine of x over cosine of x.

00:00:49.987 --> 00:00:52.404
And since it can be expressed

00:00:53.578 --> 00:00:56.538
as the quotient of two functions,

00:00:56.538 --> 00:01:00.534
we can apply the quotient
rule here to evaluate this,

00:01:00.534 --> 00:01:03.013
or to figure out what this is going to be.

00:01:03.013 --> 00:01:06.818
The quotient rule tells us
that this is going to be

00:01:06.818 --> 00:01:09.227
the derivative of the top function,

00:01:09.227 --> 00:01:13.651
which we know is cosine of
x times the bottom function

00:01:13.651 --> 00:01:17.151
which is cosine of x, so times cosine of x

00:01:18.685 --> 00:01:22.852
minus, minus the top
function, which is sine of x,

00:01:24.252 --> 00:01:28.419
sine of x, times the derivative
of the bottom function.

00:01:29.572 --> 00:01:33.547
So the derivative of cosine
of x is negative sine of x,

00:01:33.547 --> 00:01:35.317
so I can put the sine of x there,

00:01:35.317 --> 00:01:38.008
but where the negative
can just cancel that out.

00:01:38.008 --> 00:01:40.591
And it's going to be over, over

00:01:42.054 --> 00:01:44.028
the bottom function squared.

00:01:44.028 --> 00:01:45.945
So cosine squared of x.

00:01:47.345 --> 00:01:48.801
Now, what is this?

00:01:48.801 --> 00:01:52.968
Well, what we have here, this
is just a cosine squared of x,

00:01:54.299 --> 00:01:56.882
this is just sine squared of x.

00:01:58.356 --> 00:02:00.484
And we know from the Pythagorean identity,

00:02:00.484 --> 00:02:01.513
and this is really just out of,

00:02:01.513 --> 00:02:03.495
comes out of the unit circle definition,

00:02:03.495 --> 00:02:06.429
the cosine squared of x
plus sine squared of x,

00:02:06.429 --> 00:02:09.007
well that's gonna be
equal to one for any x.

00:02:09.007 --> 00:02:11.223
So all of this is equal to one.

00:02:11.223 --> 00:02:15.223
And so we end up with one
over cosine squared x,

00:02:16.775 --> 00:02:21.076
which is the same thing as,
which is the same thing as,

00:02:21.076 --> 00:02:23.326
the secant of x squared.

00:02:24.779 --> 00:02:26.760
One over cosine of x is secant,

00:02:26.760 --> 00:02:29.636
so this is just secant of x squared.

00:02:29.636 --> 00:02:30.889
So that was pretty straightforward.

00:02:30.889 --> 00:02:33.403
Now, let's just do the inverse of the,

00:02:33.403 --> 00:02:35.089
or you could say the
reciprocal, I should say,

00:02:35.089 --> 00:02:38.270
of the tangent function,
which is the cotangent.

00:02:38.270 --> 00:02:40.331
Oh, that was fun, so let's do that,

00:02:40.331 --> 00:02:41.831
d dx of cotangent,

00:02:43.685 --> 00:02:46.185
not cosine, of cotangent of x.

00:02:48.053 --> 00:02:52.220
Well, same idea, that's the
derivative with respect to x,

00:02:53.626 --> 00:02:58.166
and this time, let me make some
sufficiently large brackets.

00:02:58.166 --> 00:03:01.916
So now this is cosine of x over sine of x,

00:03:02.840 --> 00:03:04.090
over sine of x.

00:03:05.558 --> 00:03:08.523
But once again, we can use
the quotient rule here,

00:03:08.523 --> 00:03:12.157
so this is going to be the
derivative of the top function

00:03:12.157 --> 00:03:15.845
which is negative, use that magenta color.

00:03:15.845 --> 00:03:18.012
That is negative sine of x

00:03:20.921 --> 00:03:23.351
times the bottom function,

00:03:23.351 --> 00:03:25.851
so times sine of x, sine of x,

00:03:27.242 --> 00:03:28.242
minus, minus

00:03:32.172 --> 00:03:34.672
the top function, cosine of x,

00:03:35.590 --> 00:03:40.409
cosine of x, times the
derivative of the bottom function

00:03:40.409 --> 00:03:43.680
which is just going to
be another cosine of x,

00:03:43.680 --> 00:03:47.785
and then all of that over
the bottom function squared.

00:03:47.785 --> 00:03:49.785
So sine of x squared.

00:03:51.145 --> 00:03:52.631
Now what does this simplify to?

00:03:52.631 --> 00:03:56.464
Up here, let's see, this
is sine squared of x,

00:03:58.870 --> 00:04:00.337
we have a negative there,

00:04:00.337 --> 00:04:02.504
minus cosine squared of x.

00:04:03.836 --> 00:04:05.321
But we could factor out the negative

00:04:05.321 --> 00:04:08.988
and this would be
negative sine squared of x

00:04:09.877 --> 00:04:11.960
plus cosine squared of x.

00:04:13.131 --> 00:04:16.593
Well, this is just one by
the Pythagorean identity,

00:04:16.593 --> 00:04:19.662
and so this is negative
one over sine squared x,

00:04:19.662 --> 00:04:22.412
negative one over sine squared x.

00:04:23.853 --> 00:04:26.270
And that is the same thing as

00:04:27.278 --> 00:04:29.445
negative cosecant squared,

00:04:30.955 --> 00:04:33.538
I'm running out of space, of x.

00:04:35.426 --> 00:04:36.509
There you go.