Radical functions differentiation

WEBVTT
Kind: captions
Language: en

00:00:00.291 --> 00:00:01.572
- [Voiceover] Let's see if
we can take the derivative

00:00:01.572 --> 00:00:03.638
with respect to x of the fourth root

00:00:03.638 --> 00:00:07.759
of x to the third power plus
four x squared plus seven.

00:00:07.759 --> 00:00:09.209
And at first you might say, "All right,

00:00:09.209 --> 00:00:11.252
"how do I take the
derivative of the fourth root

00:00:11.252 --> 00:00:13.703
"of something, it looks like
I have a composite function,

00:00:13.703 --> 00:00:17.423
"I'm taking the fourth root
of another expression?"

00:00:17.423 --> 00:00:18.256
And you'd be right.

00:00:18.256 --> 00:00:19.711
And if you're dealing
with a composite function,

00:00:19.711 --> 00:00:22.592
the chain rule should be front of mind.

00:00:22.592 --> 00:00:24.666
But first, let's just
make this fourth root

00:00:24.666 --> 00:00:27.288
a little bit more tractable for us.

00:00:27.288 --> 00:00:29.264
And just realize that this fourth root

00:00:29.264 --> 00:00:31.274
is nothing but a fractional exponent.

00:00:31.274 --> 00:00:33.531
So this is the same
thing as the derivative

00:00:33.531 --> 00:00:37.698
with respect to x of x to
the third plus four x squared

00:00:40.164 --> 00:00:44.029
plus seven to the 1/4 power,

00:00:44.029 --> 00:00:46.362
to the 1/4 power.

00:00:47.483 --> 00:00:51.079
Now, how do we take
the derivative of this?

00:00:51.079 --> 00:00:54.501
Well, we can view this, as
I said a few seconds ago,

00:00:54.501 --> 00:00:56.407
we can view this as a composite function.

00:00:56.407 --> 00:00:57.497
What do we do first with our x?

00:00:57.497 --> 00:00:58.995
Well, we do all of this business,

00:00:58.995 --> 00:01:01.218
and we can call this u of x.

00:01:01.218 --> 00:01:02.978
And then whatever we get for u of x,

00:01:02.978 --> 00:01:05.061
we raise that to the fourth power.

00:01:05.061 --> 00:01:06.749
So the way that we would
take the derivative,

00:01:06.749 --> 00:01:08.398
we would take the derivative of this,

00:01:08.398 --> 00:01:10.146
you could view it as the outer function

00:01:10.146 --> 00:01:12.111
with respect to u of x
and then multiply that

00:01:12.111 --> 00:01:15.778
times the derivative
of u with respect to x.

00:01:17.172 --> 00:01:18.589
So let's do that.

00:01:19.677 --> 00:01:21.503
So what this is going to be,

00:01:21.503 --> 00:01:23.988
this is going to be equal to,

00:01:23.988 --> 00:01:27.583
so we're gonna take our outside function,

00:01:27.583 --> 00:01:29.893
which I'm highlighting in green now,

00:01:29.893 --> 00:01:32.267
so, or I take something to the 1/4,

00:01:32.267 --> 00:01:34.000
I'm gonna take the derivative
of that with respect

00:01:34.000 --> 00:01:36.841
to the inside, with respect to u of x.

00:01:36.841 --> 00:01:38.218
Well, I'm just gonna
use the power rule here.

00:01:38.218 --> 00:01:40.626
I'm just gonna bring that 1/4 out front,

00:01:40.626 --> 00:01:43.506
so it's gonna be 1/4 times

00:01:43.506 --> 00:01:46.006
whatever I'm taking the
derivative with respect to,

00:01:46.006 --> 00:01:49.891
to the 1/4 minus one power.

00:01:49.891 --> 00:01:52.147
Look, all I did was use
the power rule here.

00:01:52.147 --> 00:01:53.774
I didn't have an x here.

00:01:53.774 --> 00:01:56.105
Now I'm taking the derivative
with respect to u of x,

00:01:56.105 --> 00:01:59.338
with respect to this
polynomial expression here.

00:01:59.338 --> 00:02:03.261
So I could just throw the
u of x in here if I like,

00:02:03.261 --> 00:02:04.234
actually let me just do that.

00:02:04.234 --> 00:02:06.909
So, this is going to be x to the third

00:02:06.909 --> 00:02:10.278
plus four x squared plus seven.

00:02:10.278 --> 00:02:12.061
And then I wanna multiply that,

00:02:12.061 --> 00:02:12.986
and this is the chain rule.

00:02:12.986 --> 00:02:14.601
I took the derivative of the outside

00:02:14.601 --> 00:02:17.461
with respect to the inside and
then I'm gonna multiply that

00:02:17.461 --> 00:02:19.979
times the derivative of the inside.

00:02:19.979 --> 00:02:22.139
So what's the derivative of u of x?

00:02:22.139 --> 00:02:25.549
U prime of x, let's see we just gonna use

00:02:25.549 --> 00:02:26.860
the power rule a bunch of times,

00:02:26.860 --> 00:02:30.242
it's gonna be three x squared

00:02:30.242 --> 00:02:32.742
plus two times four is eight x

00:02:34.452 --> 00:02:37.703
to the two minus one is
just one power, first power,

00:02:37.703 --> 00:02:40.079
so I can just write that as eight x,

00:02:40.079 --> 00:02:41.988
and then the derivative
with respect to x of seven,

00:02:41.988 --> 00:02:44.487
well, the derivative with
respect to x of a constant

00:02:44.487 --> 00:02:45.688
is just gonna be zero.

00:02:45.688 --> 00:02:46.808
So that's u prime of x.

00:02:46.808 --> 00:02:49.016
So then I'm just gonna
multiply by u prime of x

00:02:49.016 --> 00:02:51.766
which is three x squared plus,

00:02:52.628 --> 00:02:55.354
three x squared plus eight x.

00:02:55.354 --> 00:02:59.374
And so, well, I can clean
this up a little bit,

00:02:59.374 --> 00:03:01.541
so this would be equal to,

00:03:02.599 --> 00:03:04.249
this would be equal to.

00:03:04.249 --> 00:03:07.749
Actually let me just
rewrite that exponent there.

00:03:07.749 --> 00:03:11.284
So this 1/4 minus one, I can rewrite it,

00:03:11.284 --> 00:03:14.941
1/4 minus one is negative 3/4,

00:03:14.941 --> 00:03:16.941
negative 3/4,

00:03:17.963 --> 00:03:20.463
negative 3/4 power.

00:03:21.548 --> 00:03:24.288
And you can manipulate this in
different ways, if you like,

00:03:24.288 --> 00:03:26.298
but the key is to just
recognize that this is

00:03:26.298 --> 00:03:28.086
an application of the chain rule.

00:03:28.086 --> 00:03:29.932
Derivative of the outside, well, actually,

00:03:29.932 --> 00:03:31.565
the first thing to
realize is the fourth root

00:03:31.565 --> 00:03:34.087
is the same thing as taking
something to the 1/4 power,

00:03:34.087 --> 00:03:36.830
basic exponent property, and then realize,

00:03:36.830 --> 00:03:38.602
okay, I have a composite function here.

00:03:38.602 --> 00:03:40.145
So I can take the
derivative of the outside

00:03:40.145 --> 00:03:42.391
with respect to the inside,
that's what we did here,

00:03:42.391 --> 00:03:46.949
times the derivative of the
inside with respect to x.

00:03:46.949 --> 00:03:49.372
And so if someone were to tell you,

00:03:49.372 --> 00:03:53.156
if someone were to say,
"All right, f of x,

00:03:53.156 --> 00:03:56.573
"f of x is equal to the fourth root of

00:03:58.261 --> 00:04:02.701
"x to the third plus four
x squared plus seven,"

00:04:02.701 --> 00:04:05.691
and then they said,
"Well, what is f prime of

00:04:05.691 --> 00:04:08.372
"I don't know, negative three?"

00:04:08.372 --> 00:04:11.280
Well, you would evaluate
this at negative three.

00:04:11.280 --> 00:04:12.450
So let me just do that.

00:04:12.450 --> 00:04:15.298
So it's 1/4 times,

00:04:15.298 --> 00:04:17.619
see, you have negative 27,

00:04:17.619 --> 00:04:19.579
I hope this works out reasonably well,

00:04:19.579 --> 00:04:20.996
plus 36, plus 36,

00:04:23.895 --> 00:04:27.062
plus 7 to the negative 3/4,

00:04:27.971 --> 00:04:29.554
what does this result to?

00:04:29.554 --> 00:04:32.371
This is going to be equal to,

00:04:32.371 --> 00:04:35.054
this right over here is 16, right?

00:04:35.054 --> 00:04:37.586
Negative 27 plus seven is negative 20

00:04:37.586 --> 00:04:39.503
plus 36, so this is 16.

00:04:40.456 --> 00:04:42.254
I think this is going to work out nicely.

00:04:42.254 --> 00:04:43.421
And then times

00:04:45.029 --> 00:04:48.718
three times negative, so three times nine,

00:04:48.718 --> 00:04:50.468
which is 27 minus 24.

00:04:53.088 --> 00:04:55.714
So this is going to be right over here,

00:04:55.714 --> 00:04:57.404
that is going to be three.

00:04:57.404 --> 00:04:59.991
Now, what is 16 to the negative 3/4?

00:04:59.991 --> 00:05:04.861
So this is 1/4 times,
so 16 times to the 1/4

00:05:04.861 --> 00:05:08.252
is two and then you raise that to the,

00:05:08.252 --> 00:05:11.004
let me, actually I don't
want to skip steps here.

00:05:11.004 --> 00:05:13.184
At this point we're dealing with algebra,

00:05:13.184 --> 00:05:14.731
or maybe even pre-algebra.

00:05:14.731 --> 00:05:17.731
So this is going to be times, times,

00:05:19.106 --> 00:05:21.189
16 to the 1/4,

00:05:22.355 --> 00:05:24.418
and then we're gonna raise
that to the negative three

00:05:24.418 --> 00:05:26.571
times that three out front.

00:05:26.571 --> 00:05:28.167
So we could put that three there.

00:05:28.167 --> 00:05:30.747
16 to the 1/4 is two,

00:05:30.747 --> 00:05:32.384
two to the third is eight,

00:05:32.384 --> 00:05:36.009
so two to the negative third power is 1/8,

00:05:36.009 --> 00:05:37.568
so that is 1/8.

00:05:37.568 --> 00:05:40.032
So we have 3/4 times 1/8

00:05:40.032 --> 00:05:44.115
which is equal to three over 32, 3/32.

00:05:45.450 --> 00:05:47.540
So that would be the
slope of the tangent line

00:05:47.540 --> 00:05:51.103
of the graph y is equal to f of x

00:05:51.103 --> 00:05:53.868
when x is equal to negative three.


Trig functions differentiation

WEBVTT
Kind: captions
Language: en

00:00:00.467 --> 00:00:03.603
- [Voiceover] So let's say
that we have y is equal to

00:00:03.603 --> 00:00:06.936
the secant of three pi over two minus x.

00:00:10.358 --> 00:00:12.833
And what we wanna do is,
we wanna figure out what

00:00:12.833 --> 00:00:17.265
dy/dx is, the derivative
of y with respect to x is,

00:00:17.265 --> 00:00:19.515
at x equal pi over four.

00:00:21.896 --> 00:00:23.943
Like always, pause this
video and see if you could

00:00:23.943 --> 00:00:25.852
figure it out.

00:00:25.852 --> 00:00:28.661
Well as you can see here, we
have a composite function.

00:00:28.661 --> 00:00:30.856
We're taking the secant, not just of x,

00:00:30.856 --> 00:00:33.459
but you could view this
as of another expression

00:00:33.459 --> 00:00:36.925
that x could define, or as
of a, of another function.

00:00:36.925 --> 00:00:41.692
So for example, if you call
this right over here, u of x,

00:00:41.692 --> 00:00:43.040
let's do that.

00:00:43.040 --> 00:00:47.207
So if we say u of x is equal
to three pi over two minus x,

00:00:49.011 --> 00:00:51.163
we could also figure out u prime of x

00:00:51.163 --> 00:00:53.518
is going to be equal to
derivative of three pi over two,

00:00:53.518 --> 00:00:55.473
that's just going to be a zero,

00:00:55.473 --> 00:00:58.300
derivative of minus x,
well that's just gonna be

00:00:58.300 --> 00:01:01.847
minus one, and you could just
view that as a power rule.

00:01:01.847 --> 00:01:05.712
It's one times negative one,
times x to the zero power,

00:01:05.712 --> 00:01:06.866
which is just one.

00:01:06.866 --> 00:01:07.961
So there you go.

00:01:07.961 --> 00:01:11.672
So we could view this as
the derivative of the secant

00:01:11.672 --> 00:01:14.926
with respect to u of x, that
when we take the derivative,

00:01:14.926 --> 00:01:17.730
the derivative of secant
with respect to u of x,

00:01:17.730 --> 00:01:21.397
times the derivative
of u with respect to x.

00:01:22.400 --> 00:01:23.264
And you might say,

00:01:23.264 --> 00:01:24.969
"Well, what about the
derivative of secant?"

00:01:24.969 --> 00:01:27.533
Well in other videos, we
actually prove it out,

00:01:27.533 --> 00:01:28.973
and you could actually re-derive it.

00:01:28.973 --> 00:01:31.289
Secant is just one over cosine of x,

00:01:31.289 --> 00:01:33.067
so it comes straight
out of the chain rule.

00:01:33.067 --> 00:01:37.002
So, in other videos, we prove that the,

00:01:37.002 --> 00:01:40.650
the derivative of the secant of x,

00:01:40.650 --> 00:01:44.817
of the secant of x, is equal
to, is equal to sin of x,

00:01:47.274 --> 00:01:51.901
over cosine of x, over
cosine of x squared.

00:01:51.901 --> 00:01:55.915
So if we're trying to
find the derivative of y

00:01:55.915 --> 00:01:59.438
with respect to x, well it's
going to be the derivative

00:01:59.438 --> 00:02:02.044
of secant with respect to u of x,

00:02:02.044 --> 00:02:04.984
times the derivative
of u with respect to x.

00:02:04.984 --> 00:02:05.817
So let's do that.

00:02:05.817 --> 00:02:09.489
The derivative of secant
with respect to u of x,

00:02:09.489 --> 00:02:11.560
well, instead of seeing an x everywhere,

00:02:11.560 --> 00:02:13.510
you're gonna see a u of x everywhere.

00:02:13.510 --> 00:02:15.293
So this is going to be,

00:02:15.293 --> 00:02:16.460
sin of u of x,

00:02:19.018 --> 00:02:20.185
sin of u of x,

00:02:21.669 --> 00:02:23.324
and I could, I don't have to write u of x,

00:02:23.324 --> 00:02:25.144
I could write three pi over two minus x,

00:02:25.144 --> 00:02:26.797
but I'll write u of x
right over here just to

00:02:26.797 --> 00:02:28.863
really visualize what we're doing.

00:02:28.863 --> 00:02:30.967
So sin of u of x,

00:02:30.967 --> 00:02:32.487
over,

00:02:32.487 --> 00:02:33.727
over,

00:02:33.727 --> 00:02:35.810
cosine squared of u of x,

00:02:37.242 --> 00:02:38.608
cosine squared,

00:02:38.608 --> 00:02:40.098
and we do those parentheses
in the blue color

00:02:40.098 --> 00:02:41.919
just to make sure that you identify it

00:02:41.919 --> 00:02:43.948
with the trig function.

00:02:43.948 --> 00:02:46.948
So cosine squared of u of x, u of x,

00:02:49.938 --> 00:02:53.246
so that's the derivative of
secant with respect to u of x,

00:02:53.246 --> 00:02:54.402
and then the chain rule
tells us it's gonna be

00:02:54.402 --> 00:02:57.152
that times u prime, u prime of x.

00:02:59.279 --> 00:03:02.091
So what is this going to be equal to?

00:03:02.091 --> 00:03:04.409
Well, I could just substitute back.

00:03:04.409 --> 00:03:08.295
This is going to be equal to,
I will write it like this.

00:03:08.295 --> 00:03:12.462
Sin of u of x, which is
three pi over two minus x,

00:03:13.303 --> 00:03:14.674
and I'll fill that in in a second,

00:03:14.674 --> 00:03:17.424
over cosine of u of x squared,

00:03:18.324 --> 00:03:20.668
times u prime of x,

00:03:20.668 --> 00:03:23.918
u of x is three pi over two minus x,

00:03:25.256 --> 00:03:27.423
three pi over two minus x.

00:03:28.767 --> 00:03:31.666
And then u prime of x,
we already figured out,

00:03:31.666 --> 00:03:33.566
is negative one, so I could write,

00:03:33.566 --> 00:03:36.212
times negative one, oh, yeah.

00:03:36.212 --> 00:03:37.783
Let me just leave it out there for now.

00:03:37.783 --> 00:03:39.607
I could have just put
a negative out front,

00:03:39.607 --> 00:03:41.432
but I really want you to be able to see

00:03:41.432 --> 00:03:42.898
what I'm doing here.

00:03:42.898 --> 00:03:47.161
And now we want to evaluate
at x equals pi over four.

00:03:47.161 --> 00:03:49.911
So that is equal to pi over four.

00:03:51.693 --> 00:03:52.776
Pi over four.

00:03:53.882 --> 00:03:56.037
So let's see, this is going to be,

00:03:56.037 --> 00:03:59.517
this is going to be equal to sin of,

00:03:59.517 --> 00:04:02.152
what's three pi over
two minus pi over four?

00:04:02.152 --> 00:04:03.354
I'll do that over here.

00:04:03.354 --> 00:04:04.590
So we have a common denominator,

00:04:04.590 --> 00:04:06.673
that is six pi over four,

00:04:07.812 --> 00:04:09.837
the same thing as three pi over two,

00:04:09.837 --> 00:04:11.420
minus pi over four,

00:04:13.110 --> 00:04:16.193
minus pi over four, is equal to five,

00:04:18.182 --> 00:04:19.682
five pi over four.

00:04:21.824 --> 00:04:24.574
So it's sin of five pi over four,

00:04:26.056 --> 00:04:27.556
five pi over four,

00:04:29.649 --> 00:04:33.066
over cosine squared of five pi over four,

00:04:35.363 --> 00:04:39.697
and then times negative one,
I can just put that out here.

00:04:39.697 --> 00:04:41.805
Now what is sin of five pi over four

00:04:41.805 --> 00:04:43.953
and cosine squared of five pi over four?

00:04:43.953 --> 00:04:46.729
Well, I don't have that memorized, but,

00:04:46.729 --> 00:04:48.138
let's actually draw a unit circle

00:04:48.138 --> 00:04:49.958
and we should be able to,

00:04:49.958 --> 00:04:52.482
we should be able to
figure out what that is.

00:04:52.482 --> 00:04:53.982
So, a unit circle,

00:04:55.283 --> 00:04:59.169
I try to hand draw it as best as I can.

00:04:59.169 --> 00:05:01.650
Please forgive me that
this circle does not look

00:05:01.650 --> 00:05:04.339
really like a circle, alright.

00:05:04.339 --> 00:05:05.173
Okay.

00:05:05.173 --> 00:05:07.283
So, let me just remember my angles.

00:05:07.283 --> 00:05:11.170
So, I, I in my brain, I
sometimes convert into degrees.

00:05:11.170 --> 00:05:13.420
Pi over four is 45 degrees.

00:05:15.103 --> 00:05:19.103
This is pi over two, this
is three pi over four,

00:05:19.941 --> 00:05:22.133
this is four pi over four,

00:05:22.133 --> 00:05:24.300
this is five pi over four,

00:05:25.233 --> 00:05:27.832
lands you right over there.

00:05:27.832 --> 00:05:31.727
So if you wanted to see where
you intersect the unit circle,

00:05:31.727 --> 00:05:35.310
this is at the point,
this is at the point.

00:05:36.169 --> 00:05:39.420
Your x-coordinate is negative
square root of two over two,

00:05:39.420 --> 00:05:40.865
negative square root of two over two,

00:05:40.865 --> 00:05:42.773
and your y-coordinate
is negative square root

00:05:42.773 --> 00:05:44.265
of two over two.

00:05:44.265 --> 00:05:46.044
If you are wondering how I got that,

00:05:46.044 --> 00:05:48.427
I encourage you to review the unit circle

00:05:48.427 --> 00:05:52.307
and some of the standard
angles around the unit circle.

00:05:52.307 --> 00:05:54.620
You'll see that in the
trigonometry section

00:05:54.620 --> 00:05:56.365
of Khan Academy.

00:05:56.365 --> 00:05:59.429
But this is enough for us, because the sin

00:05:59.429 --> 00:06:02.200
is the y-coordinate, it's
the y-coordinate here.

00:06:02.200 --> 00:06:04.848
So negative square root of two over two.

00:06:04.848 --> 00:06:08.456
So this is negative square
root of two over two.

00:06:08.456 --> 00:06:12.472
And then, the cosine is the x-coordinate,

00:06:12.472 --> 00:06:14.586
which is also negative
square root of two over two,

00:06:14.586 --> 00:06:15.659
it's going to be that squared.

00:06:15.659 --> 00:06:20.290
Negative square root of two
over two, we're squaring it.

00:06:20.290 --> 00:06:23.310
So if we square this, it's gonna become,

00:06:23.310 --> 00:06:26.167
this is going to become,
it's gonna become positive.

00:06:26.167 --> 00:06:30.356
And then square root
of two squared is two,

00:06:30.356 --> 00:06:33.085
and then two squared is four.

00:06:33.085 --> 00:06:35.116
So it's one half.

00:06:35.116 --> 00:06:37.885
So this is, the denominator's
equal to one half.

00:06:37.885 --> 00:06:39.086
See the numerator, this negative

00:06:39.086 --> 00:06:41.197
cancels out with that negative.

00:06:41.197 --> 00:06:43.107
And so we are left with,

00:06:43.107 --> 00:06:45.339
we deserve a little bit of a drum roll.

00:06:45.339 --> 00:06:49.318
We are left with square
root of two over two,

00:06:49.318 --> 00:06:50.479
that's the numerator,

00:06:50.479 --> 00:06:52.916
divided by one half, well
that's the same thing

00:06:52.916 --> 00:06:55.480
as multiplying by two.

00:06:55.480 --> 00:06:59.480
So we are left with
positive square root of two,

00:07:00.740 --> 00:07:03.345
is the slope of the tangent line,

00:07:03.345 --> 00:07:05.666
to the graph of y is equal to this

00:07:05.666 --> 00:07:08.312
when x is equal to pi over four.

00:07:08.312 --> 00:07:09.645
Pretty exciting.


Exponential functions differentiation

WEBVTT
Kind: captions
Language: en

00:00:00.331 --> 00:00:02.053
- [Voiceover] Let's say
that y is equal to seven

00:00:02.053 --> 00:00:05.751
to the x squared minus x power.

00:00:05.751 --> 00:00:10.176
What is the derivative
of y, derivative of y,

00:00:10.176 --> 00:00:11.676
with respect to x?

00:00:12.909 --> 00:00:14.225
And like always, pause this video

00:00:14.225 --> 00:00:16.542
and see if you can figure it out.

00:00:16.542 --> 00:00:19.766
Well, based on how this has
been color-coded ahead of time,

00:00:19.766 --> 00:00:21.413
you might immediately recognize that

00:00:21.413 --> 00:00:23.845
this is a composite function,
or it could be viewed

00:00:23.845 --> 00:00:25.675
as a composite function.

00:00:25.675 --> 00:00:29.842
If you had a v of x, which
if you had a function v of x,

00:00:31.581 --> 00:00:35.097
which is equal to seven to the xth power,

00:00:35.097 --> 00:00:37.750
and you had another function u of x,

00:00:37.750 --> 00:00:42.181
u of x which is equal
to x squared minus x,

00:00:42.181 --> 00:00:44.274
then what we have right over here,

00:00:44.274 --> 00:00:47.274
y, y is equal to seven to something,

00:00:49.605 --> 00:00:53.605
so it's equal to v of,
and it's not just v of x,

00:00:54.440 --> 00:00:56.685
it's v of u of x, instead of an x here

00:00:56.685 --> 00:00:59.167
you have the whole function u of x,

00:00:59.167 --> 00:01:00.921
x squared minus x.

00:01:00.921 --> 00:01:05.807
So, it's v of u of x and
the chain rule tells us

00:01:05.807 --> 00:01:09.234
that the derivative of
y with respect to x,

00:01:09.234 --> 00:01:11.048
and you'll see different notations here,

00:01:11.048 --> 00:01:13.240
sometimes you'll see it
written as the derivative

00:01:13.240 --> 00:01:16.907
of v with respect to
u, so v prime of u of x

00:01:18.493 --> 00:01:21.326
times the derivative
of u with respect to x,

00:01:21.326 --> 00:01:23.014
so that's one way you could do it,

00:01:23.014 --> 00:01:26.293
or you could say that this is equal to,

00:01:26.293 --> 00:01:30.907
this is equal to the
derivative, the derivative of v

00:01:30.907 --> 00:01:35.074
with respect to x, sorry,
derivative of v with respect to u,

00:01:36.200 --> 00:01:40.367
d v d u times the derivative
of u with respect to x,

00:01:44.073 --> 00:01:46.906
derivative of u with respect to x,

00:01:48.732 --> 00:01:52.081
and so either way we can
apply that right over here.

00:01:52.081 --> 00:01:55.392
So, what's the derivative
of v with respect to u?

00:01:55.392 --> 00:01:57.640
What is v prime of u of x?

00:01:57.640 --> 00:02:00.975
Well, we know, we know,
let me actually write it

00:02:00.975 --> 00:02:04.583
right over here, if v of x is
equal to seven to the x power

00:02:04.583 --> 00:02:07.952
v prime of x would be equal
to, and we've proved this

00:02:07.952 --> 00:02:11.048
in other videos where we
take derivatives exponentials

00:02:11.048 --> 00:02:14.191
of bases other than e, this
going to be the natural log

00:02:14.191 --> 00:02:17.441
of seven times seven to the x power.

00:02:19.163 --> 00:02:22.277
So, if we are taking v prime of u of x,

00:02:22.277 --> 00:02:24.696
then notice instead of an x everywhere,

00:02:24.696 --> 00:02:26.735
we're going to have a u of x everywhere.

00:02:26.735 --> 00:02:29.310
So, this right over
here, this is going to be

00:02:29.310 --> 00:02:32.643
natural log of seven times seven to the,

00:02:34.510 --> 00:02:36.021
instead of saying seven to the x power,

00:02:36.021 --> 00:02:38.426
remember we're taking v prime of u of x,

00:02:38.426 --> 00:02:42.593
so it's going to be seven to
the x squared minus x power,

00:02:44.101 --> 00:02:47.518
x squared, x squared minus x power,

00:02:48.698 --> 00:02:52.094
and then we want to multiply
that times the derivative of u

00:02:52.094 --> 00:02:53.712
with respect to x.

00:02:53.712 --> 00:02:57.333
So, u prime of x, well, that's going to be

00:02:57.333 --> 00:03:00.718
two x to the first which
is just two x minus one,

00:03:00.718 --> 00:03:04.022
so we're going to
multiply this times two x,

00:03:04.022 --> 00:03:07.439
two x minus one, so there you have it,

00:03:08.602 --> 00:03:12.312
that is the derivative
of y with respect to x.

00:03:12.312 --> 00:03:14.655
You could, we could try to simplify this

00:03:14.655 --> 00:03:17.130
or I guess re-express
it in different ways,

00:03:17.130 --> 00:03:18.909
but the main thing to realize is, look,

00:03:18.909 --> 00:03:20.870
we're just gonna take the
derivative of the seven

00:03:20.870 --> 00:03:24.385
to the this to the u of x
power with respect to u of x.

00:03:24.385 --> 00:03:26.550
So, we treat the u of x the
way that we would've treated

00:03:26.550 --> 00:03:29.691
an x right over here, so it's
gonna be natural log of seven

00:03:29.691 --> 00:03:32.306
times seven to the u of x power,

00:03:32.306 --> 00:03:34.980
we take that and multiply
that times u prime of x,

00:03:34.980 --> 00:03:36.064
and once again this is just

00:03:36.064 --> 00:03:38.814
an application of the chain rule.


Logarithmic functions differentiation

WEBVTT
Kind: captions
Language: en

00:00:00.349 --> 00:00:02.139
- [Voiceover] Let's say that Y is equal to

00:00:02.139 --> 00:00:04.659
log base four of X squared plus X.

00:00:04.659 --> 00:00:07.202
What is the derivative of Y

00:00:07.202 --> 00:00:10.452
with respect to X going to be equal to?

00:00:11.517 --> 00:00:13.222
Now you might recognize immediately that

00:00:13.222 --> 00:00:14.793
this is a composite function.

00:00:14.793 --> 00:00:17.509
We're taking the log
base four, not just of X,

00:00:17.509 --> 00:00:19.387
but we're taking that
of another expression

00:00:19.387 --> 00:00:20.577
that involves X.

00:00:20.577 --> 00:00:22.122
So we could say

00:00:22.122 --> 00:00:24.683
we could say this thing in blue

00:00:24.683 --> 00:00:25.955
that's U of X.

00:00:25.955 --> 00:00:28.041
Let me do that in blue.

00:00:28.041 --> 00:00:29.646
So this thing in blue

00:00:29.646 --> 00:00:30.659
that is U of X.

00:00:30.659 --> 00:00:33.058
U of X is equal to X squared

00:00:33.058 --> 00:00:34.187
plus X.

00:00:34.187 --> 00:00:35.751
And it's gonna be useful later on to know

00:00:35.751 --> 00:00:37.265
what U prime of X is.

00:00:37.265 --> 00:00:38.377
So that's gonna be

00:00:38.377 --> 00:00:39.809
I'm just gonna use the power rule here

00:00:39.809 --> 00:00:42.001
so two X plus one

00:00:42.001 --> 00:00:43.784
I brought that two out front

00:00:43.784 --> 00:00:45.188
and decremented the exponent.

00:00:45.188 --> 00:00:48.412
Derivative with respect to X of X is one.

00:00:48.412 --> 00:00:49.985
And we can say the

00:00:49.985 --> 00:00:52.688
log base four of this stuff

00:00:52.688 --> 00:00:55.299
well we could call that a function V.

00:00:55.299 --> 00:00:56.549
We can say V of

00:00:57.485 --> 00:00:59.177
well if we said V of X

00:00:59.177 --> 00:01:01.427
this would be log base four

00:01:02.545 --> 00:01:03.778
of X.

00:01:03.778 --> 00:01:06.028
And then we've shown in other videos

00:01:06.028 --> 00:01:07.445
that V prime of X

00:01:08.685 --> 00:01:10.882
is, we're gonna be very
similar that if this

00:01:10.882 --> 00:01:13.188
was log base E, or natural log,

00:01:13.188 --> 00:01:14.728
except we're going to scale it.

00:01:14.728 --> 00:01:15.894
So it's going to be

00:01:15.894 --> 00:01:17.467
one over

00:01:17.467 --> 00:01:19.384
one over log base four.

00:01:21.462 --> 00:01:23.395
Sorry, one over

00:01:23.395 --> 00:01:25.166
the natural log.

00:01:25.166 --> 00:01:27.622
The natural log of four

00:01:27.622 --> 00:01:28.455
times X.

00:01:29.629 --> 00:01:32.100
If this was V of X, if V of X was just

00:01:32.100 --> 00:01:34.501
natural log of X, our
derivative would be one over X.

00:01:34.501 --> 00:01:36.192
But since it's log base four

00:01:36.192 --> 00:01:37.778
and this comes straight
out of the change of base

00:01:37.778 --> 00:01:39.028
formulas that you might have seen.

00:01:39.028 --> 00:01:41.026
And we have a video where we show this.

00:01:41.026 --> 00:01:42.757
But we just scale it in the denominator

00:01:42.757 --> 00:01:43.979
with this natural log of four.

00:01:43.979 --> 00:01:45.695
You think of scaling the whole expression

00:01:45.695 --> 00:01:48.179
by one over the natural log of four.

00:01:48.179 --> 00:01:51.160
But we can now use this
information because Y

00:01:51.160 --> 00:01:53.077
this Y can be viewed as

00:01:54.308 --> 00:01:55.141
V of

00:01:56.054 --> 00:01:57.673
V of.

00:01:57.673 --> 00:02:00.575
Remember, V is the log
base four of something.

00:02:00.575 --> 00:02:02.215
But it's not V of X.

00:02:02.215 --> 00:02:03.749
We don't have just an X here.

00:02:03.749 --> 00:02:05.757
We have the whole expression

00:02:05.757 --> 00:02:07.371
that defines U of X.

00:02:07.371 --> 00:02:09.891
We have U of X right there.

00:02:09.891 --> 00:02:11.840
And let me draw a little line here

00:02:11.840 --> 00:02:15.377
so that we don't get
those two sides confused.

00:02:15.377 --> 00:02:17.036
And so we know from the chain rule

00:02:17.036 --> 00:02:19.781
the derivative Y with respect to X.

00:02:19.781 --> 00:02:21.901
This is going to be

00:02:21.901 --> 00:02:23.736
this is going to be the derivative of V

00:02:23.736 --> 00:02:25.910
with respect to U.

00:02:25.910 --> 00:02:28.454
Or we could call that V prime.

00:02:28.454 --> 00:02:29.954
V prime of U of X.

00:02:31.412 --> 00:02:32.912
V prime of U of X.

00:02:34.700 --> 00:02:36.981
Let me do the U of X in blue.

00:02:36.981 --> 00:02:38.398
V prime of U of X

00:02:40.030 --> 00:02:41.613
times U prime of X.

00:02:43.443 --> 00:02:45.318
Well, what is V prime of U of X?

00:02:45.318 --> 00:02:46.900
We know what V prime of X is.

00:02:46.900 --> 00:02:48.680
If we want to know what V prime of U of X

00:02:48.680 --> 00:02:50.427
we would just replace wherever we see an X

00:02:50.427 --> 00:02:51.715
with a U of X.

00:02:51.715 --> 00:02:54.315
So, this is going to be equal to

00:02:54.315 --> 00:02:56.662
V prime of U X, U of X.

00:02:56.662 --> 00:02:58.463
And you just do is you take the derivative

00:02:58.463 --> 00:03:01.391
of the green function with
respect to the blue function.

00:03:01.391 --> 00:03:03.741
So it's going to be one over

00:03:03.741 --> 00:03:05.509
the natural log of four.

00:03:05.509 --> 00:03:07.015
The natural log of four.

00:03:07.015 --> 00:03:09.291
Times, instead of putting an X there

00:03:09.291 --> 00:03:11.757
it would be times U of X.

00:03:11.757 --> 00:03:12.840
Times U of X.

00:03:14.397 --> 00:03:16.320
And of course, that whole thing

00:03:16.320 --> 00:03:17.903
times U prime of X.

00:03:19.695 --> 00:03:22.208
And so, and I'm doing
more steps just hopefully

00:03:22.208 --> 00:03:23.943
so it's clearer what I'm doing here.

00:03:23.943 --> 00:03:25.528
So this is one over

00:03:25.528 --> 00:03:27.998
the natural log of four.

00:03:27.998 --> 00:03:30.248
U of X is X squared plus X.

00:03:31.219 --> 00:03:32.612
So

00:03:32.612 --> 00:03:34.029
X squared plus X.

00:03:35.072 --> 00:03:36.079
And then we're gonna multiply that

00:03:36.079 --> 00:03:37.246
times U prime of X.

00:03:37.246 --> 00:03:39.385
So times two X plus one.

00:03:39.385 --> 00:03:41.162
And so we can just rewrite this as

00:03:41.162 --> 00:03:42.745
two X plus one over

00:03:45.296 --> 00:03:46.129
over

00:03:47.747 --> 00:03:49.017
over

00:03:49.017 --> 00:03:50.745
the natural log of four.

00:03:50.745 --> 00:03:52.863
The natural log of four

00:03:52.863 --> 00:03:54.780
times X squared plus X.

00:03:55.617 --> 00:03:56.867
Times X squared

00:03:58.962 --> 00:03:59.795
plus X.

00:03:59.795 --> 00:04:00.867
And we're done, and we could distribute

00:04:00.867 --> 00:04:02.143
this natural log of four

00:04:02.143 --> 00:04:03.650
if we found that interesting.

00:04:03.650 --> 00:04:06.010
But, we have just found
the derivative of Y

00:04:06.010 --> 00:04:07.593
with respect to X.


Derivatives of inverse functions: from table

WEBVTT
Kind: captions
Language: en

00:00:00.375 --> 00:00:04.171
- [Voiceover] Let G and
H be inverse functions.

00:00:04.171 --> 00:00:05.880
So let's just remind
ourselves what it means

00:00:05.880 --> 00:00:08.631
for them to be inverse functions.

00:00:08.631 --> 00:00:13.035
That means that if I
have two sets of numbers,

00:00:13.035 --> 00:00:15.305
let's say one set right over there,

00:00:15.305 --> 00:00:17.949
that's another set right over there,

00:00:17.949 --> 00:00:21.607
and if we view that first
set as the domain of G,

00:00:21.607 --> 00:00:25.078
so if you start with
some X right over here,

00:00:25.078 --> 00:00:28.706
G is going to map from
that X to another value,

00:00:28.706 --> 00:00:30.956
which we would call G of X.

00:00:32.778 --> 00:00:34.953
That's what the function G does.

00:00:34.953 --> 00:00:39.279
Now if H is the inverse of
G and frankly vice versa,

00:00:39.279 --> 00:00:43.898
then H could go from that
point G of X back to X.

00:00:43.898 --> 00:00:45.794
So H would do this.

00:00:45.794 --> 00:00:49.746
H would get us back to our original value.

00:00:49.746 --> 00:00:52.657
So that's what the function H would do

00:00:52.657 --> 00:00:55.354
and so we could view this
point right over here,

00:00:55.354 --> 00:00:58.852
we could view it as X, so that is X,

00:00:58.852 --> 00:01:02.964
but we could also view it as H of G of X.

00:01:02.964 --> 00:01:06.381
So we could also view this as H of G of X

00:01:08.945 --> 00:01:10.654
and I did all of that so we can

00:01:10.654 --> 00:01:13.281
really feel good about this idea.

00:01:13.281 --> 00:01:16.659
If someone tells you that G
and H are inverse functions,

00:01:16.659 --> 00:01:19.409
that means that H of G of X is X.

00:01:22.266 --> 00:01:23.266
H of G of X.

00:01:25.524 --> 00:01:27.691
H of G of X is equal to X.

00:01:29.263 --> 00:01:31.212
Or you could have gone
the other way around.

00:01:31.212 --> 00:01:34.176
You could have started with,

00:01:34.176 --> 00:01:36.018
well you could have done
it multiple different ways,

00:01:36.018 --> 00:01:37.807
but also G of H of X.

00:01:37.807 --> 00:01:39.223
I could have just swapped
these letters here.

00:01:39.223 --> 00:01:41.840
The letters H and G
are somewhat arbitrary.

00:01:41.840 --> 00:01:46.032
So you could have also
said that G of H of X

00:01:46.032 --> 00:01:47.199
is equal to X.

00:01:48.569 --> 00:01:50.986
So G of H of X is equal to X.

00:01:52.441 --> 00:01:53.909
And then they give us some information.

00:01:53.909 --> 00:01:58.076
The following table lists a few
values of G, H, and G prime.

00:01:59.277 --> 00:02:03.202
Alright, so they want us to
evaluate H prime of three.

00:02:03.202 --> 00:02:05.231
They don't even give us H prime of three.

00:02:05.231 --> 00:02:06.064
How do we figure it out?

00:02:06.064 --> 00:02:09.046
They gave us G prime and H and G.

00:02:09.046 --> 00:02:10.889
How do we figure this out?

00:02:10.889 --> 00:02:13.425
Well here we're going to
actually derive something

00:02:13.425 --> 00:02:15.722
based on the chain rule and
this isn't the type of problem

00:02:15.722 --> 00:02:18.855
that you'll see a lot
of, but it is interesting

00:02:18.855 --> 00:02:20.658
so we're going to work through
it and there's a chance

00:02:20.658 --> 00:02:23.382
that you might see it
in your calculus class.

00:02:23.382 --> 00:02:27.841
So let's start with either one
of these expressions up here.

00:02:27.841 --> 00:02:30.758
So let's start with the expression,

00:02:32.354 --> 00:02:35.879
well let's start with,
let's do this one over here.

00:02:35.879 --> 00:02:39.212
So if we have G of H of X is equal to X,

00:02:43.195 --> 00:02:46.026
so we put that H of X back there,

00:02:46.026 --> 00:02:50.193
which is by definition true
if G and H are inverses.

00:02:51.099 --> 00:02:53.583
Well now let's take the
derivative of both sides of this.

00:02:53.583 --> 00:02:54.624
So let's take the derivative

00:02:54.624 --> 00:02:58.015
with respect to X of both sides,

00:02:58.015 --> 00:03:00.846
derivative with respect to X,

00:03:00.846 --> 00:03:03.730
and on the left-hand side well
we just apply the chain rule.

00:03:03.730 --> 00:03:06.397
This would be G prime of H of X,

00:03:09.631 --> 00:03:11.131
G prime of H of X,

00:03:14.037 --> 00:03:15.620
times H prime of X.

00:03:17.482 --> 00:03:19.565
That's just the chain
rule right over there

00:03:19.565 --> 00:03:21.567
and then that would be equal to,

00:03:21.567 --> 00:03:24.291
what's the derivative
with respect to X of X?

00:03:24.291 --> 00:03:27.068
Well that's just going to be equal to one.

00:03:27.068 --> 00:03:28.590
So now it's interesting.

00:03:28.590 --> 00:03:31.821
We need to figure out
what H prime of three is.

00:03:31.821 --> 00:03:33.931
We can figure out what H of three is

00:03:33.931 --> 00:03:35.846
and then we can use that to figure out

00:03:35.846 --> 00:03:40.453
what G prime of whatever
G prime of H of three is

00:03:40.453 --> 00:03:42.669
and so we should be able
to figure out H prime of X

00:03:42.669 --> 00:03:45.019
or we could just rewrite it this way.

00:03:45.019 --> 00:03:48.936
We could rewrite that H
prime of X is equal to,

00:03:51.417 --> 00:03:54.667
is equal to one over G prime of H of X.

00:03:57.986 --> 00:04:00.817
Now in some circles,
they might encourage you

00:04:00.817 --> 00:04:02.472
to memorize this and maybe for the sake

00:04:02.472 --> 00:04:04.311
of doing this exercise on Khan Academy

00:04:04.311 --> 00:04:05.566
you might want to memorize it,

00:04:05.566 --> 00:04:08.450
but I'll tell you 20 years after I took,

00:04:08.450 --> 00:04:10.479
almost 25 years after I took calculus,

00:04:10.479 --> 00:04:13.684
this is not something that I
retain in my long-term memory,

00:04:13.684 --> 00:04:16.087
but I did retain that
you can derive this from

00:04:16.087 --> 00:04:19.665
just what the definition of
inverse functions actually are.

00:04:19.665 --> 00:04:20.760
But we can use this now

00:04:20.760 --> 00:04:23.724
if we want to figure out
what H prime of three is.

00:04:23.724 --> 00:04:27.756
H prime of three is going to be equal to

00:04:27.756 --> 00:04:30.339
one over G prime of H of three,

00:04:35.206 --> 00:04:37.770
which I'm guessing that
they have given us.

00:04:37.770 --> 00:04:41.241
So H of three, when X is three, H is four.

00:04:41.241 --> 00:04:43.484
So that is H of three there.

00:04:43.484 --> 00:04:44.980
So H of three is four.

00:04:44.980 --> 00:04:47.223
So now we just have to
figure out G prime of four.

00:04:47.223 --> 00:04:49.573
Well lucky for us, they have given us

00:04:49.573 --> 00:04:53.740
when X is equal to four,
G prime is equal to 1/2.

00:04:55.180 --> 00:04:58.465
So G prime of four is equal to 1/2.

00:04:58.465 --> 00:05:02.310
So H prime of three is
equal to one over 1/2.

00:05:02.310 --> 00:05:03.511
So one over 1/2,

00:05:03.511 --> 00:05:06.369
one divided by 1/2 is the
same thing as one times two.

00:05:06.369 --> 00:05:10.036
So this is all equal
to two and we are done.


Derivatives of inverse functions: from equation

WEBVTT
Kind: captions
Language: en

00:00:00.480 --> 00:00:02.905
- [Voiceover] Let F of X be
equal to one half X to the third

00:00:02.905 --> 00:00:04.948
plus three X minus four.

00:00:04.948 --> 00:00:07.603
Let H be the inverse of F.

00:00:07.603 --> 00:00:10.435
Notice that F of negative
two is equal to negative 14.

00:00:10.435 --> 00:00:13.908
And then they're asking us
what is H prime of negative 14?

00:00:13.908 --> 00:00:16.729
And if you're not familiar
with the how functions

00:00:16.729 --> 00:00:20.488
and their derivatives
relate to their inverses

00:00:20.488 --> 00:00:21.960
and the derivatives of the
inverse, well this will

00:00:21.960 --> 00:00:23.556
seem like a very hard thing to do.

00:00:23.556 --> 00:00:25.752
Because if you're attempting
to take the inverse of F

00:00:25.752 --> 00:00:27.592
to figure out what H is

00:00:27.592 --> 00:00:29.656
well, it's tough to find, to take

00:00:29.656 --> 00:00:32.692
to figure out the
inverse of a third degree

00:00:32.692 --> 00:00:36.424
a third degree polynomial
defined function like this.

00:00:36.424 --> 00:00:37.687
So, the key

00:00:37.687 --> 00:00:40.991
the key, I guess property to realize,

00:00:40.991 --> 00:00:42.489
or the key truth to realize

00:00:42.489 --> 00:00:44.406
if F and H are inverses

00:00:46.101 --> 00:00:47.518
then H prime of X

00:00:48.447 --> 00:00:49.447
H prime of X

00:00:50.385 --> 00:00:52.855
is going to be equal to

00:00:52.855 --> 00:00:55.296
is going to be equal to one over

00:00:55.296 --> 00:00:56.796
F prime of H of X.

00:00:59.336 --> 00:01:01.586
One over F prime of H of X.

00:01:02.528 --> 00:01:04.350
And you could now use this

00:01:04.350 --> 00:01:06.552
in order to figure out what H prime of

00:01:06.552 --> 00:01:08.537
negative 14 is.

00:01:08.537 --> 00:01:10.195
Now I know what some of you are thinking,

00:01:10.195 --> 00:01:11.752
because it's exactly
what I would be thinking

00:01:11.752 --> 00:01:14.169
if someone just sprung this on me is

00:01:14.169 --> 00:01:16.303
where does this come from?

00:01:16.303 --> 00:01:18.870
And I would tell you,
this comes straight out

00:01:18.870 --> 00:01:20.432
of the chain rule.

00:01:20.432 --> 00:01:24.183
We know that if a function and its inverse

00:01:24.183 --> 00:01:26.417
we know that if we have a
function and its inverse

00:01:26.417 --> 00:01:30.000
that F of F of the
inverse of our function.

00:01:31.122 --> 00:01:32.372
So F of H of X.

00:01:33.566 --> 00:01:34.706
F of H of X.

00:01:34.706 --> 00:01:37.571
We know that this is going to be to X.

00:01:37.571 --> 00:01:39.400
This literally, this is

00:01:39.400 --> 00:01:41.847
comes out of them being
each others inverses.

00:01:41.847 --> 00:01:43.769
We could have also said H of F of X

00:01:43.769 --> 00:01:45.743
will also be equal to X.

00:01:45.743 --> 00:01:47.103
Remember, F is going to map

00:01:47.103 --> 00:01:50.700
or H is going to map
from some X to H of X.

00:01:50.700 --> 00:01:53.333
And then F is going to map
back to that original X.

00:01:53.333 --> 00:01:55.286
That's what inverses do.

00:01:55.286 --> 00:01:57.525
So that's because they are inverses.

00:01:57.525 --> 00:01:59.700
This is by definition,
this is what inverses

00:01:59.700 --> 00:02:00.759
do to each other.

00:02:00.759 --> 00:02:02.232
And then if you took the derivative of

00:02:02.232 --> 00:02:03.719
both sides of this

00:02:03.719 --> 00:02:05.066
what would you get?

00:02:05.066 --> 00:02:05.949
Let me do that.

00:02:05.949 --> 00:02:07.447
So if we took the derivative

00:02:07.447 --> 00:02:08.853
of the both sides of this

00:02:08.853 --> 00:02:12.004
D D X on the left hand side.

00:02:12.004 --> 00:02:14.800
D D X on the right hand side.

00:02:14.800 --> 00:02:16.188
I think you see where this is going.

00:02:16.188 --> 00:02:18.090
You are essentially gonna
get a version of that.

00:02:18.090 --> 00:02:19.768
The left hand side used the chain rule.

00:02:19.768 --> 00:02:22.935
You're going to get F prime of H of X.

00:02:24.315 --> 00:02:25.732
F prime of H of X

00:02:28.012 --> 00:02:29.996
times H prime of X

00:02:29.996 --> 00:02:31.436
comes straight out of the chain rule

00:02:31.436 --> 00:02:34.797
is equal to, is equal
to the derivative of X

00:02:34.797 --> 00:02:36.533
is just going to be equal to one.

00:02:36.533 --> 00:02:38.567
And then you derive,
you divide both sides by

00:02:38.567 --> 00:02:39.828
F prime of H of X

00:02:39.828 --> 00:02:42.037
and you get our original property there.

00:02:42.037 --> 00:02:43.421
So now with that out of the way

00:02:43.421 --> 00:02:45.506
let's just actually apply this.

00:02:45.506 --> 00:02:48.662
So, we want to evaluate H prime of 14.

00:02:48.662 --> 00:02:51.488
Or sorry, H prime of negative 14.

00:02:51.488 --> 00:02:53.986
Is going to be equal to

00:02:53.986 --> 00:02:55.569
one over F prime of

00:02:58.355 --> 00:02:59.772
H of negative 14.

00:03:01.473 --> 00:03:02.890
H of negative 14.

00:03:04.815 --> 00:03:08.002
Now had they given us H of negative 14.

00:03:08.002 --> 00:03:09.564
But they didn't give it to us explicitly,

00:03:09.564 --> 00:03:12.099
we have to remember that
F and H are inverses

00:03:12.099 --> 00:03:13.003
of each other.

00:03:13.003 --> 00:03:15.274
So F of negative two is negative 14.

00:03:15.274 --> 00:03:18.063
Well, H is gonna go from
the other way around.

00:03:18.063 --> 00:03:19.857
If you input negative 14 into H

00:03:19.857 --> 00:03:21.667
you're going to get negative two.

00:03:21.667 --> 00:03:23.250
So H of negative 14

00:03:24.383 --> 00:03:26.416
well, this is going to be equal to

00:03:26.416 --> 00:03:27.467
negative two.

00:03:27.467 --> 00:03:29.889
Once again, they are
inverses of each other.

00:03:29.889 --> 00:03:32.682
So H of negative 14 is equal to

00:03:32.682 --> 00:03:34.594
negative negative two.

00:03:34.594 --> 00:03:36.621
And once again, I just
swapped these two around.

00:03:36.621 --> 00:03:38.816
That's what the inverse function will do.

00:03:38.816 --> 00:03:40.782
If you're wrapping from if F goes from

00:03:40.782 --> 00:03:41.989
negative two to negative 14

00:03:41.989 --> 00:03:44.033
H is going to go from negative 14

00:03:44.033 --> 00:03:45.783
back to negative two.

00:03:47.216 --> 00:03:50.998
So now we want to evaluate
F prime of negative two.

00:03:50.998 --> 00:03:54.313
Well, let's figure out
what F prime of X is.

00:03:54.313 --> 00:03:56.646
So, F prime of X is equal to

00:03:58.389 --> 00:04:01.932
remember the power rule,
so three times one half

00:04:01.932 --> 00:04:03.182
is three halves

00:04:04.150 --> 00:04:06.929
times X to the three minus one power

00:04:06.929 --> 00:04:08.550
which is the second power.

00:04:08.550 --> 00:04:09.876
Plus the derivative of three X

00:04:09.876 --> 00:04:13.276
with respect to X which
that's just going to be three.

00:04:13.276 --> 00:04:15.545
And you could do that,
it's just the power rule.

00:04:15.545 --> 00:04:16.933
But this was X to the first power,

00:04:16.933 --> 00:04:18.161
one times three,

00:04:18.161 --> 00:04:19.504
X to the zero power,

00:04:19.504 --> 00:04:21.204
but X to the zero is just one

00:04:21.204 --> 00:04:22.148
so you're just left with three.

00:04:22.148 --> 00:04:24.328
And derivative of a concept
that's just gonna be zero.

00:04:24.328 --> 00:04:25.800
So that's F prime of X.

00:04:25.800 --> 00:04:28.967
So F prime of, F prime of negative two

00:04:30.341 --> 00:04:33.379
is going to be three halves

00:04:33.379 --> 00:04:36.763
times negative two squared
is four, positive four.

00:04:36.763 --> 00:04:38.124
So plus three.

00:04:38.124 --> 00:04:40.392
So, this is going to be equal to

00:04:40.392 --> 00:04:43.333
two times three plus three.

00:04:43.333 --> 00:04:46.307
So, six plus three is equal to nine.

00:04:46.307 --> 00:04:49.010
So this denominator here is
going to be equal to nine.

00:04:49.010 --> 00:04:52.515
So this whole thing is
equal to one over nine.

00:04:52.515 --> 00:04:55.993
So this involved, this
was something you're not

00:04:55.993 --> 00:04:56.974
going to see every day.

00:04:56.974 --> 00:04:59.655
This isn't that typical
problem in your calculus class.

00:04:59.655 --> 00:05:01.405
But it's interesting.


Derivatives expressed as limits

WEBVTT
Kind: captions
Language: en

00:00:00.494 --> 00:00:01.416
- [Voiceover] Let's see
if we can find the limit

00:00:01.416 --> 00:00:05.230
as h approaches zero of
five log of two plus h

00:00:05.230 --> 00:00:07.063
minus five log of two,

00:00:08.025 --> 00:00:10.198
all of that over h.

00:00:10.198 --> 00:00:11.603
And I'll give you a little bit of a hint

00:00:11.603 --> 00:00:12.728
because I know you're
about to pause the video

00:00:12.728 --> 00:00:14.112
and try to work through it.

00:00:14.112 --> 00:00:16.294
Think of your derivative properties,

00:00:16.294 --> 00:00:19.187
especially the derivatives
of logarithmic functions,

00:00:19.187 --> 00:00:21.315
especially logarithmic
functions in this case

00:00:21.315 --> 00:00:22.672
with a base 10.

00:00:22.672 --> 00:00:24.444
If someone just writes
log without the base,

00:00:24.444 --> 00:00:28.027
you can just assume that that
is a 10 right over there.

00:00:28.027 --> 00:00:30.725
So pause the video, and see
if you can work through it.

00:00:30.725 --> 00:00:33.816
Alright, so the key here
is to remember that if

00:00:33.816 --> 00:00:35.899
I have, if I have f of x,

00:00:37.419 --> 00:00:38.603
let me do it over here,

00:00:38.603 --> 00:00:39.585
I'll do it over here.

00:00:39.585 --> 00:00:40.418
F of x

00:00:41.553 --> 00:00:43.247
and I want to find

00:00:43.247 --> 00:00:44.164
f prime of,

00:00:45.530 --> 00:00:48.280
let's say f prime of some number,

00:00:49.271 --> 00:00:50.571
let's say a,

00:00:50.571 --> 00:00:52.422
this is going to be equal to

00:00:52.422 --> 00:00:54.505
the limit as x, or sorry,

00:00:56.005 --> 00:00:58.604
as h approaches zero is one
way of thinking about it,

00:00:58.604 --> 00:01:01.771
as h approaches zero of f of a plus h,

00:01:03.369 --> 00:01:04.452
minus f of a,

00:01:05.771 --> 00:01:07.779
all of that over h.

00:01:07.779 --> 00:01:10.595
So this looks pretty close
to that limit definition,

00:01:10.595 --> 00:01:12.465
except we have these fives here.

00:01:12.465 --> 00:01:15.542
But lucky for us we can
factor out those fives,

00:01:15.542 --> 00:01:17.156
and we could factor them out,

00:01:17.156 --> 00:01:19.658
we could factor them out out front here,

00:01:19.658 --> 00:01:22.157
but if you just have a scale
or times the expression,

00:01:22.157 --> 00:01:23.636
we know from our limit properties

00:01:23.636 --> 00:01:26.312
we can actually take those
out of the limit themselves.

00:01:26.312 --> 00:01:27.191
So let's do that,

00:01:27.191 --> 00:01:28.733
let's take both of these fives,

00:01:28.733 --> 00:01:29.856
and factor them out,

00:01:29.856 --> 00:01:31.490
and so this whole thing
is going to simplify to

00:01:31.490 --> 00:01:33.157
five times the limit

00:01:34.522 --> 00:01:38.022
as h approaches zero of log of two plus h,

00:01:41.984 --> 00:01:43.984
minus, minus log of two,

00:01:47.772 --> 00:01:49.355
all of that over h.

00:01:50.923 --> 00:01:54.348
Now, you might recognize
what we have in yellow here,

00:01:54.348 --> 00:01:55.229
cause think about it,

00:01:55.229 --> 00:01:56.652
what this is,

00:01:56.652 --> 00:01:59.211
if we had f of x is equal to

00:01:59.211 --> 00:02:00.044
log of x,

00:02:02.539 --> 00:02:06.524
and we wanted to know what f prime of,

00:02:06.524 --> 00:02:09.206
well actually let's say f prime of two is,

00:02:09.206 --> 00:02:11.923
well this would be the limit

00:02:11.923 --> 00:02:14.423
as h approaches zero of log of

00:02:16.274 --> 00:02:17.948
two plus h,

00:02:17.948 --> 00:02:18.865
two plus h,

00:02:19.700 --> 00:02:21.216
minus log of two,

00:02:21.216 --> 00:02:23.205
minus log of two,

00:02:23.205 --> 00:02:25.429
all of that over h.

00:02:25.429 --> 00:02:27.064
So this is really just a,

00:02:27.064 --> 00:02:28.009
what we see there,

00:02:28.009 --> 00:02:29.978
this by definition

00:02:29.978 --> 00:02:32.183
this right over here is,

00:02:32.183 --> 00:02:33.683
is f prime of two.

00:02:35.117 --> 00:02:36.632
If f of x is log of x,

00:02:36.632 --> 00:02:38.522
this is f prime of two,

00:02:38.522 --> 00:02:39.772
f prime of two.

00:02:40.671 --> 00:02:41.850
So can we figure that out?

00:02:41.850 --> 00:02:44.213
If f of x is log of x,

00:02:44.213 --> 00:02:46.083
what is f prime of x?

00:02:46.083 --> 00:02:49.095
F prime of x, we don't need
to use the limit definition.

00:02:49.095 --> 00:02:50.296
In fact the limit definition is quite

00:02:50.296 --> 00:02:52.285
hard to evaluate this limit,

00:02:52.285 --> 00:02:53.958
but we know how to take the derivative

00:02:53.958 --> 00:02:56.223
of logarithmic functions.

00:02:56.223 --> 00:02:57.975
So f prime of x

00:02:57.975 --> 00:02:59.471
is going to be equal to

00:02:59.471 --> 00:03:03.155
one over the natural log of our base,

00:03:03.155 --> 00:03:04.847
our base here we already
talked about that,

00:03:04.847 --> 00:03:06.205
that is 10.

00:03:06.205 --> 00:03:07.996
So one over natural log of 10,

00:03:07.996 --> 00:03:08.829
times,

00:03:10.438 --> 00:03:11.688
times, times x.

00:03:13.588 --> 00:03:15.144
If this was a natural log,

00:03:15.144 --> 00:03:17.231
well, then, this would be one over

00:03:17.231 --> 00:03:18.845
natural log of e times x,

00:03:18.845 --> 00:03:20.046
natural log of e is just one,

00:03:20.046 --> 00:03:21.247
so that's where you get the one over x,

00:03:21.247 --> 00:03:22.468
but if you have any other base

00:03:22.468 --> 00:03:23.866
you put the natural log of that base

00:03:23.866 --> 00:03:25.618
right over here in the denominator.

00:03:25.618 --> 00:03:27.469
So what is f prime of two?

00:03:27.469 --> 00:03:28.886
F prime of two is

00:03:28.886 --> 00:03:31.702
one over the natural log of 10,

00:03:31.702 --> 00:03:32.535
times two.

00:03:34.458 --> 00:03:36.368
So this whole thing has simplified,

00:03:36.368 --> 00:03:40.365
this whole thing is equal
to five times this business.

00:03:40.365 --> 00:03:41.447
So I could actually just write it as

00:03:41.447 --> 00:03:43.447
it's equal to five over,

00:03:44.755 --> 00:03:47.925
five over the natural log of 10,

00:03:47.925 --> 00:03:49.599
natural log of 10,

00:03:49.599 --> 00:03:50.740
times two,

00:03:50.740 --> 00:03:53.477
I could have written it
as two natural log of 10s.

00:03:53.477 --> 00:03:55.722
The key here for this type of exercise,

00:03:55.722 --> 00:03:56.785
you might be a little let me see

00:03:56.785 --> 00:03:57.710
if I can evaluate this limit?

00:03:57.710 --> 00:03:59.817
You're like whoa, this looks
a lot like the derivative

00:03:59.817 --> 00:04:01.372
of a logarithmic function,

00:04:01.372 --> 00:04:04.129
especially the derivative
when x is equal to two,

00:04:04.129 --> 00:04:06.019
if we could just factor these fives out.

00:04:06.019 --> 00:04:07.023
So you factor out the fives,

00:04:07.023 --> 00:04:09.071
and say hey, this is,
this is the derivative

00:04:09.071 --> 00:04:12.488
of log of x when x is equal to two,

00:04:13.973 --> 00:04:16.966
and so we know how to take
the derivative of log of x.

00:04:16.966 --> 00:04:20.096
If you don't know we have
videos where we prove this.

00:04:20.096 --> 00:04:22.342
Where you take the
derivatives of logarithms

00:04:22.342 --> 00:04:24.526
with bases other than e,

00:04:24.526 --> 00:04:26.554
and you just use that to actually

00:04:26.554 --> 00:04:28.169
and you find the derivative,

00:04:28.169 --> 00:04:29.193
and you evaluate it at two,

00:04:29.193 --> 00:04:30.943
and then you're done!


Implicit differentiation, product and chain rules at once

WEBVTT
Kind: captions
Language: en

00:00:00.361 --> 00:00:02.806
- [Voiceover] Let's say y is equal to

00:00:02.806 --> 00:00:06.223
the natural log of x to the xth power,

00:00:08.494 --> 00:00:10.029
and what we wanna do, we wanna find

00:00:10.029 --> 00:00:13.401
the derivative of y with respect to x.

00:00:13.401 --> 00:00:14.777
So I encourage you to pause this video

00:00:14.777 --> 00:00:17.332
and see if you can do it.

00:00:17.332 --> 00:00:18.744
So, when you first try to tackle this,

00:00:18.744 --> 00:00:20.225
this is a little bit daunting.

00:00:20.225 --> 00:00:22.866
We know how to take the
derivative of constants

00:00:22.866 --> 00:00:26.184
to some x power, but how
do we take a derivative

00:00:26.184 --> 00:00:28.689
of some time of a function, in this case,

00:00:28.689 --> 00:00:31.356
a natural log, to the xth power?

00:00:32.217 --> 00:00:33.785
And the answer here is to use some of

00:00:33.785 --> 00:00:36.264
our logarithmic properties,
and then we're going to do

00:00:36.264 --> 00:00:38.767
a little bit of implicit differentiation.

00:00:38.767 --> 00:00:40.866
So the first thing that we wanna do,

00:00:40.866 --> 00:00:43.319
actually, let me rewrite this
with a little bit of space,

00:00:43.319 --> 00:00:47.486
so, this was a natural
log of x to the xth power.

00:00:50.934 --> 00:00:53.796
So the first thing, I
wanna get rid of this x

00:00:53.796 --> 00:00:55.925
as an exponential, and I wanna be able

00:00:55.925 --> 00:00:57.954
to apply the product rule somehow.

00:00:57.954 --> 00:00:59.777
The way we're gonna do that is by taking

00:00:59.777 --> 00:01:02.393
the natural log of both sides.

00:01:02.393 --> 00:01:04.613
So take the natural log of both sides,

00:01:04.613 --> 00:01:06.405
and you might say, well,
why is that helpful?

00:01:06.405 --> 00:01:08.367
Well, if I'm taking the natural log

00:01:08.367 --> 00:01:11.697
of something to an exponent,
well, this is the same thing,

00:01:11.697 --> 00:01:13.089
actually, let me write this down

00:01:13.089 --> 00:01:15.313
as a property that you
may or may not remember

00:01:15.313 --> 00:01:16.942
from your logarithmic properties,

00:01:16.942 --> 00:01:19.167
so if I have, I could write log,

00:01:19.167 --> 00:01:20.778
or I'll just write natural log,

00:01:20.778 --> 00:01:25.349
of, if I have natural
log of a to the b power,

00:01:25.349 --> 00:01:30.299
this is the same thing as b
times the natural log of a.

00:01:30.299 --> 00:01:32.676
It's just a standard logarithmic property.

00:01:32.676 --> 00:01:35.038
And so, by taking the
natural log of both sides,

00:01:35.038 --> 00:01:37.861
this exponent can now become out front

00:01:37.861 --> 00:01:40.052
and scale the natural log function.

00:01:40.052 --> 00:01:44.219
So this exponent now, we
can bring that out front,

00:01:45.482 --> 00:01:47.589
and let's just rewrite everything.

00:01:47.589 --> 00:01:50.089
So we get the natural log of y

00:01:51.936 --> 00:01:56.757
is equal to, so then let
me put that in parentheses,

00:01:56.757 --> 00:01:58.581
so it's the natural log of y,

00:01:58.581 --> 00:02:01.581
is equal to x, that x is in blue,

00:02:04.150 --> 00:02:08.317
x times the natural log times
the natural log, of, sorry,

00:02:11.710 --> 00:02:15.710
x times the natural log
of the natural log of x.

00:02:19.803 --> 00:02:23.136
The natural log of the natural log of x.

00:02:24.426 --> 00:02:26.027
So, there you have it.

00:02:26.027 --> 00:02:27.551
By just taking the
natural log of both sides

00:02:27.551 --> 00:02:29.318
and using this logarithmic property,

00:02:29.318 --> 00:02:31.908
we were able to get that.

00:02:31.908 --> 00:02:33.256
Now you're saying, well, gee,

00:02:33.256 --> 00:02:37.347
how is this actually
going to be useful for us?

00:02:37.347 --> 00:02:38.968
Well now, we can implicitly take

00:02:38.968 --> 00:02:41.683
the derivative of both sides of this.

00:02:41.683 --> 00:02:44.445
And actually, let me scoot this over

00:02:44.445 --> 00:02:46.751
to the right a little
bit, just so that we can,

00:02:46.751 --> 00:02:50.084
I have space for my derivative operator.

00:02:51.131 --> 00:02:54.781
So there you go, scooted that over.

00:02:54.781 --> 00:02:56.090
And so now, let's take the derivative

00:02:56.090 --> 00:02:58.908
with respect to x of both sides.

00:02:58.908 --> 00:03:02.584
So let me, so I'm gonna
take the derivative

00:03:02.584 --> 00:03:06.133
with respect to x of the left-hand side

00:03:06.133 --> 00:03:08.383
and of the right-hand side,

00:03:09.473 --> 00:03:12.447
and of the right-hand side.

00:03:12.447 --> 00:03:15.571
Now, on the left-hand
side, this is going to be,

00:03:15.571 --> 00:03:17.563
essentially, an application
of the chain rule.

00:03:17.563 --> 00:03:18.877
When you learn implicit differentiation,

00:03:18.877 --> 00:03:20.599
it's really just application
of the chain rule.

00:03:20.599 --> 00:03:22.244
It's the derivative of
the outside function

00:03:22.244 --> 00:03:23.948
with respect to to the inside function,

00:03:23.948 --> 00:03:26.290
so the natural log of y with respect to y,

00:03:26.290 --> 00:03:30.465
the derivative of that is
just going to be one over y,

00:03:30.465 --> 00:03:33.298
one over y times the derivative of

00:03:35.587 --> 00:03:37.943
the inside function with respect to x.

00:03:37.943 --> 00:03:39.276
So dy dx, dy dx.

00:03:43.713 --> 00:03:46.227
That is going to be equal to,

00:03:46.227 --> 00:03:49.277
well this is going to get
interesting a little bit.

00:03:49.277 --> 00:03:54.037
Actually, let me do some stuff
on the side a little bit.

00:03:54.037 --> 00:03:56.543
Let me just, the first
thing we wanna do here

00:03:56.543 --> 00:03:58.396
is just apply the product rule.

00:03:58.396 --> 00:04:01.036
So it's the derivative
of the first expression,

00:04:01.036 --> 00:04:04.684
so it's just going to
be 1 times the second,

00:04:04.684 --> 00:04:06.074
I guess you'd say, function,

00:04:06.074 --> 00:04:10.093
so times the natural log
of the natural log of x,

00:04:10.093 --> 00:04:12.932
natural log of x, and then

00:04:12.932 --> 00:04:15.599
plus the first function, just x,

00:04:19.099 --> 00:04:22.361
times the derivative
of the second function.

00:04:22.361 --> 00:04:25.856
Times the derivative
of the second function.

00:04:25.856 --> 00:04:27.786
What's the derivative of the natural log

00:04:27.786 --> 00:04:29.515
of the natural log of x?

00:04:29.515 --> 00:04:31.060
Let's do that separately.

00:04:31.060 --> 00:04:35.057
So if I have, if I'm trying
to take the derivative,

00:04:35.057 --> 00:04:38.474
with respect to to x, of the natural log,

00:04:40.391 --> 00:04:44.307
the natural log of the natural log of x,

00:04:44.307 --> 00:04:46.661
of the natural log of x,

00:04:46.661 --> 00:04:48.995
well, here, again, I can
apply the chain rule.

00:04:48.995 --> 00:04:50.906
The derivative of that magenta function

00:04:50.906 --> 00:04:52.897
with respect to the inside function,

00:04:52.897 --> 00:04:57.228
that is going to be one
over the natural log of x,

00:04:57.228 --> 00:04:59.956
and then times the derivative
of the inside function

00:04:59.956 --> 00:05:03.696
with respect to x, so times one over x.

00:05:03.696 --> 00:05:07.863
So this is equal to one
over x natural log of x.

00:05:10.009 --> 00:05:13.624
So the derivative of this
second function right over here

00:05:13.624 --> 00:05:16.574
is one over x natural log of x,

00:05:16.574 --> 00:05:19.157
one over x natural log of x.

00:05:21.393 --> 00:05:24.429
Let's see, that x and that x cancels out,

00:05:24.429 --> 00:05:28.596
and so we are left with, we
are left with one over y,

00:05:29.667 --> 00:05:32.794
and I'll just write all of
this in this blue color,

00:05:32.794 --> 00:05:36.044
so one over y times the derivative of y

00:05:37.021 --> 00:05:39.604
with respect to x, is equal to,

00:05:40.493 --> 00:05:44.304
see this is just the natural
log of the natural log of x,

00:05:44.304 --> 00:05:47.637
the natural log of the natural log of x,

00:05:49.153 --> 00:05:52.508
plus one over the natural log of x,

00:05:52.508 --> 00:05:55.008
one over the natural log of x.

00:05:57.827 --> 00:05:58.908
And now, to solve for the derivative,

00:05:58.908 --> 00:06:02.797
we can multiply both sides
by y, so let's do that.

00:06:02.797 --> 00:06:06.133
So we're gonna multiply that side by y,

00:06:06.133 --> 00:06:10.389
and we're gonna multiply
this side times y,

00:06:10.389 --> 00:06:12.622
and what are we going to get?

00:06:12.622 --> 00:06:15.256
Well, on the left-hand side,
that's why we multiplied by y,

00:06:15.256 --> 00:06:18.786
we just have the derivative
y with respect to x,

00:06:18.786 --> 00:06:21.743
derivative of y with respect to x,

00:06:21.743 --> 00:06:24.826
is equal to, well, y is our original,

00:06:28.609 --> 00:06:31.756
is all this, is our
original thing that we had.

00:06:31.756 --> 00:06:33.366
Y was equal to the natural log,

00:06:33.366 --> 00:06:35.170
let me rewrite it over here.

00:06:35.170 --> 00:06:39.337
Y was equal to the natural
log of x to the xth power.

00:06:41.221 --> 00:06:43.728
So this is, we're essentially
multiplying both sides

00:06:43.728 --> 00:06:47.645
times the natural log
of x to the xth power.

00:06:49.718 --> 00:06:53.885
So this gonna get a little
bit, a little bit messy here.

00:06:55.462 --> 00:06:59.103
So we could just write it
the way I wrote it just now,

00:06:59.103 --> 00:07:01.380
without it being distributed,

00:07:01.380 --> 00:07:02.770
actually, let me just leave it like that.

00:07:02.770 --> 00:07:04.612
So it's going to be, and so we deserve

00:07:04.612 --> 00:07:06.606
a drum roll right now, 'cause
this is quite involved,

00:07:06.606 --> 00:07:09.856
the natural log of the natural log of x

00:07:11.354 --> 00:07:14.271
plus one over the natural log of x,

00:07:16.337 --> 00:07:20.504
all of that times the natural
log of x to the xth power.

00:07:23.684 --> 00:07:25.217
So that was quite involved.

00:07:25.217 --> 00:07:27.739
And if someone said well,
what is the derivative of y

00:07:27.739 --> 00:07:30.540
when x is equal to e?

00:07:30.540 --> 00:07:32.026
So if someone says what is this equal to

00:07:32.026 --> 00:07:33.781
when x is equal to e, well,
we could evaluate this

00:07:33.781 --> 00:07:34.849
when x is equal to e.

00:07:34.849 --> 00:07:38.797
This would be the, and I
just made that up just now,

00:07:38.797 --> 00:07:40.405
so if like the original
question wasn't just

00:07:40.405 --> 00:07:42.745
what is dy dx, if they said

00:07:42.745 --> 00:07:45.912
what is dy dx when x is equal to e.

00:07:49.777 --> 00:07:51.388
If that was the original question,

00:07:51.388 --> 00:07:53.410
then we could evaluate it.

00:07:53.410 --> 00:07:57.020
So where we just replace
all these with e's,

00:07:57.020 --> 00:08:00.853
so it'd be an e there,
an e there, an e there,

00:08:02.054 --> 00:08:04.762
and an e there, and I
just picked the value e

00:08:04.762 --> 00:08:06.183
'cause it's easy to evaluate.

00:08:06.183 --> 00:08:08.184
So that natural log of e is 1,

00:08:08.184 --> 00:08:10.790
natural log of 1 e to
the zeroth power is 1,

00:08:10.790 --> 00:08:12.458
so all of that just becomes 0.

00:08:12.458 --> 00:08:14.625
The natural log of e is 1,

00:08:15.629 --> 00:08:18.424
so this whole expression right over here

00:08:18.424 --> 00:08:20.342
becomes zero plus one over one,

00:08:20.342 --> 00:08:22.242
so this becomes one, and then

00:08:22.242 --> 00:08:26.409
the natural log of e is,
the natural log of e is one,

00:08:27.468 --> 00:08:29.759
and you're gonna have
one to the eth power,

00:08:29.759 --> 00:08:32.175
well you could raise one to any power,

00:08:32.175 --> 00:08:33.572
and you're just get one.

00:08:33.572 --> 00:08:36.497
So it's one times one is equal to one.

00:08:36.497 --> 00:08:37.368
So I just thought it would be fun

00:08:37.368 --> 00:08:39.722
to try to evaluate that at a value

00:08:39.722 --> 00:08:42.139
that would be somewhat clean.


Chain rule with the power rule

WEBVTT
Kind: captions
Language: en

00:00:00.310 --> 00:00:01.672
- [Voiceover] So we've got the function

00:00:01.672 --> 00:00:02.721
f of x is equal to two x to the third plus

00:00:02.721 --> 00:00:03.554
five x squared minus seven.

00:00:03.554 --> 00:00:05.875
All of that to the eighth power.

00:00:05.875 --> 00:00:08.277
And we want to find the derivative of our

00:00:08.277 --> 00:00:10.609
function f with respect to x.

00:00:10.609 --> 00:00:12.527
Now the key here is to
realize that this function

00:00:12.527 --> 00:00:16.285
can be viewed as a
composition of two functions.

00:00:16.285 --> 00:00:17.587
How do we do that?

00:00:17.587 --> 00:00:19.077
Well let me diagram it out.

00:00:19.077 --> 00:00:21.662
So let's say we want to start with,

00:00:21.662 --> 00:00:23.830
and I'll do it down here
so we have some space.

00:00:23.830 --> 00:00:25.641
So we're gonna start with an x and

00:00:25.641 --> 00:00:27.469
what's the first thing that we would do?

00:00:27.469 --> 00:00:29.491
If you were just trying to
evaluate it given some x?

00:00:29.491 --> 00:00:31.480
Well, the first thing
you would take two times

00:00:31.480 --> 00:00:33.590
that x to the third power
plus five times that

00:00:33.590 --> 00:00:35.899
x squared and then minus seven.

00:00:35.899 --> 00:00:39.149
So, what if we imagined a function here

00:00:40.099 --> 00:00:41.859
that just did that first part.

00:00:41.859 --> 00:00:45.639
That just evaluated two x to the third

00:00:45.639 --> 00:00:48.749
plus five x squared
minus seven for your x.

00:00:48.749 --> 00:00:50.770
So let's call that the function u.

00:00:50.770 --> 00:00:52.571
So whatever you input
into that function u,

00:00:52.571 --> 00:00:54.701
you're gonna get two times that input

00:00:54.701 --> 00:00:57.800
to the third power, plus
five times that input

00:00:57.800 --> 00:01:00.550
to the second power, minus seven.

00:01:01.399 --> 00:01:03.230
And so when you do that,
when you input with an x,

00:01:03.230 --> 00:01:04.759
what do you output?

00:01:04.759 --> 00:01:07.180
What do you output here?

00:01:07.180 --> 00:01:10.180
Well, you're going to output u of x,

00:01:11.120 --> 00:01:15.189
which is equal to two
x to the third power,

00:01:15.189 --> 00:01:17.921
plus five x squared minus seven.

00:01:17.921 --> 00:01:19.600
And I'll do it all in one color just so

00:01:19.600 --> 00:01:21.120
I don't have to keep changing colors.

00:01:21.120 --> 00:01:23.129
So two x to the third power, plus

00:01:23.129 --> 00:01:26.099
five x squared, minus seven.

00:01:26.099 --> 00:01:27.599
That is u of x.

00:01:27.599 --> 00:01:29.099
Now what's the next thing you're gonna do?

00:01:29.099 --> 00:01:30.299
You're not done evaluating f of x yet.

00:01:30.299 --> 00:01:31.969
You would then take that value and then

00:01:31.969 --> 00:01:33.329
input into another function.

00:01:33.329 --> 00:01:36.239
You would then take the
eight power of that value.

00:01:36.239 --> 00:01:38.910
So then, we will take that and

00:01:38.910 --> 00:01:41.319
input it into another function.

00:01:41.319 --> 00:01:43.841
Let's call that function v.

00:01:43.841 --> 00:01:45.599
And that function,
whatever input you give it,

00:01:45.599 --> 00:01:47.140
and I'm using these squares just to say

00:01:47.140 --> 00:01:48.954
whatever input goes into that function.

00:01:48.954 --> 00:01:50.969
You're going to take
it to the eighth power.

00:01:50.969 --> 00:01:52.700
And so in this case, what do you get?

00:01:52.700 --> 00:01:54.289
What do you end up with?

00:01:54.289 --> 00:01:57.122
Well, you end up with v of u of x.

00:01:58.701 --> 00:01:59.701
V of u of x.

00:02:01.869 --> 00:02:06.450
Or you could view this as
v of two x to the third,

00:02:06.450 --> 00:02:09.450
plus five x squared, minus seven.

00:02:11.259 --> 00:02:15.240
Or, you could view this
as two x to the third,

00:02:15.240 --> 00:02:18.720
plus five x squared, minus seven.

00:02:18.720 --> 00:02:21.069
All of that to the eighth power.

00:02:21.069 --> 00:02:24.630
All of that to the eighth power
and that's what f of x is.

00:02:24.630 --> 00:02:26.590
So as we just saw, f of x can be viewed

00:02:26.590 --> 00:02:29.090
as the composition of v and u.

00:02:32.148 --> 00:02:33.398
This is f of x.

00:02:34.757 --> 00:02:37.037
So, if we write f of x.

00:02:37.037 --> 00:02:39.277
If we write f of x

00:02:39.277 --> 00:02:41.527
being equal to v of u of x,

00:02:46.170 --> 00:02:49.969
then we see very clearly the
chain rule is very useful here.

00:02:49.969 --> 00:02:52.771
The chain rule tells us that f prime of x

00:02:52.771 --> 00:02:56.938
is going to be the derivative
of v, with respect to u.

00:02:59.087 --> 00:03:02.712
So it's going to be v prime of, not x,

00:03:02.712 --> 00:03:04.158
but v prime of u of x.

00:03:04.158 --> 00:03:07.537
The derivative of v, with respect to u,

00:03:07.537 --> 00:03:11.617
times the derivative of
u, with respect to x.

00:03:11.617 --> 00:03:13.637
So u prime of x.

00:03:13.637 --> 00:03:16.304
So we know a few things already.

00:03:17.289 --> 00:03:20.347
So let's, let me just write
things down very clearly.

00:03:20.347 --> 00:03:23.708
So we know that u of x is equal
to two x to the third power,

00:03:23.708 --> 00:03:26.908
plus five x squared, minus seven.

00:03:26.908 --> 00:03:28.947
What is u prime of x?

00:03:28.947 --> 00:03:30.797
Here we're just going
to use some derivative

00:03:30.797 --> 00:03:32.407
properties and the power rule.

00:03:32.407 --> 00:03:34.138
Three times two is six x.

00:03:34.138 --> 00:03:37.108
Three minus one is two, six x squared.

00:03:37.108 --> 00:03:38.858
Two times five is 10.

00:03:40.048 --> 00:03:42.018
Take one off that exponent, it's gonna be

00:03:42.018 --> 00:03:44.087
10 x to the first power, or just 10 x.

00:03:44.087 --> 00:03:46.158
And the derivative of a
constant is just zero,

00:03:46.158 --> 00:03:47.737
so we can just ignore that.

00:03:47.737 --> 00:03:49.597
So that's u prime of x.

00:03:49.597 --> 00:03:53.868
Now, we know that v, if
we inputted an x into v,

00:03:53.868 --> 00:03:58.438
so v of x would be equal
to x to the eighth power.

00:03:58.438 --> 00:04:01.778
V prime of x, well we just
use the power rule again.

00:04:01.778 --> 00:04:04.627
That's eight x to the seventh power.

00:04:04.627 --> 00:04:06.710
And so v prime of u of x,

00:04:08.368 --> 00:04:10.917
so if you were to input
u of x into v prime,

00:04:10.917 --> 00:04:13.627
well it's going to be equal to

00:04:13.627 --> 00:04:17.210
eight times u of x to the seventh power.

00:04:21.366 --> 00:04:22.984
Whatever you input into v prime,

00:04:22.984 --> 00:04:24.774
you're gonna take it to the seventh power

00:04:24.774 --> 00:04:26.614
and multiply it by eight.

00:04:26.614 --> 00:04:29.864
So u of x and that's the same thing.

00:04:31.431 --> 00:04:33.331
That is the same thing.

00:04:33.331 --> 00:04:37.451
This is equal to eight times
this entire expression.

00:04:37.451 --> 00:04:40.868
U of x is two x to the third power,

00:04:41.982 --> 00:04:45.343
plus five x squared, minus seven.

00:04:45.343 --> 00:04:46.992
So there you have it.

00:04:46.992 --> 00:04:48.075
F prime of x.

00:04:49.284 --> 00:04:50.951
It is equal to this,

00:04:52.052 --> 00:04:54.152
which we just figured
out v prime of u of x is

00:04:54.152 --> 00:04:55.822
all of this business.

00:04:55.822 --> 00:04:59.989
So it's equal to eight,
times two x to the third,

00:05:01.064 --> 00:05:04.114
plus five x squared, minus seven.

00:05:04.114 --> 00:05:06.174
All of that to the seventh power,

00:05:06.174 --> 00:05:08.174
times u prime of x.

00:05:08.174 --> 00:05:09.484
U prime of x, we figured out is that.

00:05:09.484 --> 00:05:12.651
So times six x squared, plus ten x.

00:05:15.931 --> 00:05:17.502
Now, as you get more
practice with the chain rule,

00:05:17.502 --> 00:05:19.711
you'll recognize this faster

00:05:19.711 --> 00:05:21.583
and actually you won't have
to right all of this down,

00:05:21.583 --> 00:05:22.951
you'll be able to do a
lot of it in your head.

00:05:22.951 --> 00:05:24.871
You'll say, okay, I'm
gonna take the derivative

00:05:24.871 --> 00:05:27.110
of the outside function, the
blue function you could say,

00:05:27.110 --> 00:05:29.200
with respect to what I have in the inside.

00:05:29.200 --> 00:05:31.391
So, if I was taking the
derivative of x to the eighth,

00:05:31.391 --> 00:05:33.811
it would be eight x to the seventh.

00:05:33.811 --> 00:05:35.431
That's with respect to x.

00:05:35.431 --> 00:05:37.111
But if I'm taking the derivative of this,

00:05:37.111 --> 00:05:38.971
with respect to the inside, well, where I

00:05:38.971 --> 00:05:40.821
had the x's before, I would
just have this u of x.

00:05:40.821 --> 00:05:44.631
So it's gonna be eight times
this to the seventh power.

00:05:44.631 --> 00:05:47.780
And I multiply that times
the derivative of the inside.

00:05:47.780 --> 00:05:50.780
Which is six x squared plus 10 x.


Derivative of log_x (for any positive base aÃÂ1)

WEBVTT
Kind: captions
Language: en

00:00:00.117 --> 00:00:01.594
- [Voiceover] We know
from previous videos,

00:00:01.594 --> 00:00:04.575
that the derivative with respect to X

00:00:04.575 --> 00:00:08.408
of the natural log of
X, is equal to 1 over X.

00:00:12.350 --> 00:00:14.718
What I want to do in this
video is use that knowledge

00:00:14.718 --> 00:00:17.031
that we've seen in other
videos to figure out

00:00:17.031 --> 00:00:19.929
what the derivative with respect to X is

00:00:19.929 --> 00:00:23.412
of a logarithm of an arbitrary base.

00:00:23.412 --> 00:00:27.162
So I'm just gonna call
that log, base A of X.

00:00:30.517 --> 00:00:32.328
So how do we figure this out?

00:00:32.328 --> 00:00:35.198
Well, the key thing is, is
what you might be familiar with

00:00:35.198 --> 00:00:37.566
from your algebra or your
pre calculus classes,

00:00:37.566 --> 00:00:40.186
which is having a change of base.

00:00:40.186 --> 00:00:43.306
So if I have some, I'll do it over here,

00:00:43.306 --> 00:00:47.597
log, base A of B, and
I wanted to change it

00:00:47.597 --> 00:00:49.436
to a different base, let's
say I wanna change it

00:00:49.436 --> 00:00:53.603
to base C, this is the same
thing as log, base C of B

00:00:56.430 --> 00:00:58.763
divided by log, base C of A.

00:01:04.065 --> 00:01:08.050
Log, base C of B, divided
by log, base C of A.

00:01:08.050 --> 00:01:09.777
This is a really useful
thing if you've never

00:01:09.777 --> 00:01:11.700
seen it before, you now have just seen it,

00:01:11.700 --> 00:01:13.873
this change of base, and we prove it

00:01:13.873 --> 00:01:16.520
in other videos on Khan Academy.

00:01:16.520 --> 00:01:18.499
But it's really useful
because, for example,

00:01:18.499 --> 00:01:21.006
your calculator has a log button.

00:01:21.006 --> 00:01:24.740
The log on your calculator
is log, base 10.

00:01:24.740 --> 00:01:28.557
So if you press 100 into your calculator

00:01:28.557 --> 00:01:31.734
and press log, you will get a 2 there.

00:01:31.734 --> 00:01:33.935
So whenever you just see log of 100,

00:01:33.935 --> 00:01:37.195
it's implicitly base 10,
and you also have a button

00:01:37.195 --> 00:01:40.372
for natural log, which is log, base E.

00:01:40.372 --> 00:01:44.205
Natural log of X is equal
to log, base E of X.

00:01:46.028 --> 00:01:48.146
But sometimes, you wanna
find all sorts of different

00:01:48.146 --> 00:01:52.298
base logarithms and this is how you do it.

00:01:52.298 --> 00:01:54.722
So if you're using your
calculator and you wanted to find

00:01:54.722 --> 00:01:58.055
what log, base 3 of 8 is, you would say,

00:02:01.493 --> 00:02:05.979
you would type in your
calculator log of 8 and log of 3.

00:02:05.979 --> 00:02:09.248
Or, let me write it this way, and log of 3

00:02:09.248 --> 00:02:11.700
where both of these
are implicitly base 10,

00:02:11.700 --> 00:02:13.121
and you'd get the same value if you did

00:02:13.121 --> 00:02:16.871
natural log of 8 divided
by natural log of 3.

00:02:17.942 --> 00:02:19.920
Which you might also
have on your calculator.

00:02:19.920 --> 00:02:23.988
And what we're gonna do
in this video is leverage

00:02:23.988 --> 00:02:26.412
the natural log because we
know what the derivative

00:02:26.412 --> 00:02:28.586
of the natural log is.

00:02:28.586 --> 00:02:31.929
So this derivative is the
same thing as the derivative

00:02:31.929 --> 00:02:33.679
with respect to X of.

00:02:35.273 --> 00:02:38.394
Well log, base A of X, can be rewritten as

00:02:38.394 --> 00:02:41.644
natural log of X over natural log of A.

00:02:43.994 --> 00:02:46.056
And now natural log of
A, that's just a number.

00:02:46.056 --> 00:02:50.626
I could rewrite this as,
let me write it this way.

00:02:50.626 --> 00:02:54.709
One over natural log of
A times natural log of X.

00:02:56.450 --> 00:02:58.094
And what's the derivative of that?

00:02:58.094 --> 00:03:00.128
We could just take the constant out.

00:03:00.128 --> 00:03:02.357
One over natural log of
A, that's just a number.

00:03:02.357 --> 00:03:06.435
So we're gonna get 1
over the natural log of A

00:03:06.435 --> 00:03:10.602
times the derivative with
respect to X of natural log of X.

00:03:15.017 --> 00:03:16.684
Of natural log of X.

00:03:18.090 --> 00:03:20.264
Which we already know is 1 over X.

00:03:20.264 --> 00:03:23.134
So this thing right
over here, is 1 over X.

00:03:23.134 --> 00:03:27.301
So what we get is 1 over
natural log of A times 1 over X.

00:03:29.236 --> 00:03:33.403
Which we could write as, 1
over natural log of A times X.

00:03:40.409 --> 00:03:42.388
Which is a really useful thing to know.

00:03:42.388 --> 00:03:46.317
So now, we could take
all sorts of derivatives.

00:03:46.317 --> 00:03:50.484
So if I were to tell you F of
X is equal to log, base 7 of X

00:03:56.654 --> 00:04:01.084
well now we can say well F
prime of X is going to be

00:04:01.084 --> 00:04:04.084
1 over the natural log of 7 times X.

00:04:07.047 --> 00:04:11.199
If we had a constant out
front, if we had for example,

00:04:11.199 --> 00:04:12.648
G of X.

00:04:12.648 --> 00:04:16.815
G of X is equal to negative
3 times log, base, I know.

00:04:18.499 --> 00:04:20.394
Log, base pi.

00:04:20.394 --> 00:04:21.564
Pi is a number.

00:04:21.564 --> 00:04:25.731
Log, base pi of X, well G
prime of X would be equal to

00:04:28.335 --> 00:04:29.252
1 over, oh.

00:04:30.509 --> 00:04:32.013
Let me be careful, I have
this constant out here.

00:04:32.013 --> 00:04:35.914
So it'd be negative 3 over,
it's just that negative 3,

00:04:35.914 --> 00:04:38.164
over the natural log of pi.

00:04:40.261 --> 00:04:42.128
This is the natural log of this number.

00:04:42.128 --> 00:04:42.961
Times X.

00:04:44.190 --> 00:04:46.879
So hopefully, that gives
you a hang of things.


Parentheses

WEBVTT
Kind: captions
Language: en

00:00:00.637 --> 00:00:02.353
- [Voiceover] Hey grammarians, hey Paige.

00:00:02.353 --> 00:00:03.186
- [Voiceover] Hi David.

00:00:03.186 --> 00:00:04.908
- [Voiceover] So today, we're
gonna talk about parentheses.

00:00:04.908 --> 00:00:07.243
So, before we get into
what parentheses do,

00:00:07.243 --> 00:00:10.748
I would like to talk very
briefly about the word origin

00:00:10.748 --> 00:00:12.676
of parentheses or parenthesis,
- [Voiceover] Okay.

00:00:12.676 --> 00:00:16.843
because it comes from Greek,
so para means besides,

00:00:18.659 --> 00:00:21.571
and thesis means placing, right?

00:00:21.571 --> 00:00:24.654
So it's placed beside, approximately,

00:00:26.279 --> 00:00:27.636
and how we use parentheses is

00:00:27.636 --> 00:00:30.969
we kinda put 'em to the side for little,

00:00:33.489 --> 00:00:37.166
what are called asides in
writing, little interruptions.

00:00:37.166 --> 00:00:41.807
So let's lay out the
functions of parentheses.

00:00:41.807 --> 00:00:43.644
And I should say, the singular form

00:00:43.644 --> 00:00:46.144
of parentheses is parenthesis.

00:00:52.363 --> 00:00:54.492
That's just one of these.

00:00:54.492 --> 00:00:57.409
Two of 'em is parentheses, like so.

00:00:58.862 --> 00:01:03.480
So Paige, what is a parenthesis,
and what does it do?

00:01:03.480 --> 00:01:08.199
- [Voiceover] So a parenthesis
is a piece of punctuation

00:01:08.199 --> 00:01:11.410
that has kind of a lot
of different functions.

00:01:11.410 --> 00:01:15.110
It can indicate remarks
by the writer of a text.

00:01:15.110 --> 00:01:16.254
- [Voiceover] Okay.

00:01:16.254 --> 00:01:20.995
- [Voiceover] Or specify a
definition or a reference.

00:01:20.995 --> 00:01:24.583
It can also show
interruptions by an audience,

00:01:24.583 --> 00:01:27.763
but overall, the parenthesis separates

00:01:27.763 --> 00:01:31.337
a piece of technically
unnecessary information,

00:01:31.337 --> 00:01:35.419
or what we called an aside,
from the rest of the sentence.

00:01:35.419 --> 00:01:38.378
- [Voiceover] So it separates
inessential information.

00:01:38.378 --> 00:01:40.744
- [Voiceover] Right, you can take out

00:01:40.744 --> 00:01:42.215
what's in the parentheses,

00:01:42.215 --> 00:01:44.069
and the sentence will still make sense.

00:01:44.069 --> 00:01:45.920
- [Voiceover] All right, so
let's take these one at a time.

00:01:45.920 --> 00:01:47.295
So here we've got something like,

00:01:47.295 --> 00:01:50.097
the cookie, which was
still warm, was delicious.

00:01:50.097 --> 00:01:51.220
Now we could, conceivably,

00:01:51.220 --> 00:01:53.983
take that out of the sentence, right?

00:01:53.983 --> 00:01:57.314
'Cause the sentence is basically,
the cookie was delicious.

00:01:57.314 --> 00:01:59.542
But if we wanted to

00:01:59.542 --> 00:02:02.834
add an additional remark by a writer,

00:02:02.834 --> 00:02:06.432
which is whoever ate
the cookie, namely me,

00:02:06.432 --> 00:02:09.506
we would put in this parenthetical remark.

00:02:09.506 --> 00:02:12.147
- [Voiceover] Right, it's
some extra information.

00:02:12.147 --> 00:02:14.064
- [Voiceover] And when
I say parenthetical,

00:02:14.064 --> 00:02:17.698
I'm talking about this little aside here.

00:02:17.698 --> 00:02:19.909
That's what I mean by a parenthetical.

00:02:19.909 --> 00:02:22.377
So this is what's called
a parenthetical aside.

00:02:22.377 --> 00:02:24.443
So the second way in
which you use parentheses

00:02:24.443 --> 00:02:26.865
is to specify a definition or reference.

00:02:26.865 --> 00:02:28.967
So let's say I wanted to quote

00:02:28.967 --> 00:02:32.018
Paige's as yet unwritten autobiography.

00:02:32.018 --> 00:02:33.367
- [Voiceover] Okay.

00:02:33.367 --> 00:02:36.187
- [Voiceover] So, what I'm doing here

00:02:36.187 --> 00:02:38.930
is I'm using this parenthetical aside

00:02:38.930 --> 00:02:41.847
to cite where in her book she said,

00:02:42.742 --> 00:02:43.901
"Oh no, a tiger!"

00:02:43.901 --> 00:02:46.172
So let's say this is, what I'm doing here,

00:02:46.172 --> 00:02:47.275
this sentence is like an excerpt

00:02:47.275 --> 00:02:50.007
from some essay that I'm writing

00:02:50.007 --> 00:02:52.226
about Paige's life and times,

00:02:52.226 --> 00:02:54.270
which I've italicized
here because it is a book.

00:02:54.270 --> 00:02:55.261
- [Voiceover] Right.

00:02:55.261 --> 00:02:57.220
- [Voiceover] So what I'm
doin' here is I have the quote,

00:02:57.220 --> 00:03:01.569
and then, in order to say where
that reference comes from,

00:03:01.569 --> 00:03:04.378
I give the author's last name,

00:03:04.378 --> 00:03:06.752
the name of the book, and then p. 38,

00:03:06.752 --> 00:03:09.888
and all of that is inside
parentheses, like so.

00:03:09.888 --> 00:03:12.798
That's the second use of the parenthesis.

00:03:12.798 --> 00:03:15.913
So let's say you're taking
down a transcript of a speech.

00:03:15.913 --> 00:03:17.189
You're typing up someone's speech,

00:03:17.189 --> 00:03:18.854
and they start coughing, and it interrupts

00:03:18.854 --> 00:03:20.811
the way they're giving a speech.

00:03:20.811 --> 00:03:23.040
Let's say I was trying
to render the sentence,

00:03:23.040 --> 00:03:27.244
"Furthermore, (coughs) I
must firmly state that."

00:03:27.244 --> 00:03:30.117
- [Voiceover] Right, so you
can include the coughing

00:03:30.117 --> 00:03:31.219
in the sentence
- [Voiceover] Right.

00:03:31.219 --> 00:03:34.632
when you're writing it
down or transcribing it.

00:03:34.632 --> 00:03:37.121
Like, if we think this is
important enough to note,

00:03:37.121 --> 00:03:39.015
we can just sort of say, voop, voop,

00:03:39.015 --> 00:03:41.234
yup, coughing happened here.

00:03:41.234 --> 00:03:42.510
- [Voiceover] I see.

00:03:42.510 --> 00:03:45.003
So those are the uses of the parenthesis.

00:03:45.003 --> 00:03:48.033
You can indicate remarks by the writer,

00:03:48.033 --> 00:03:49.635
as in our first example,

00:03:49.635 --> 00:03:52.218
or specify a definition or a reference,

00:03:52.218 --> 00:03:54.416
which we did here with
the reference to my book,

00:03:54.416 --> 00:03:57.161
or you can show an interruption in speech.

00:03:57.161 --> 00:03:58.360
That's it!

00:03:58.360 --> 00:03:59.842
- [Voiceover] Sweet!

00:03:59.842 --> 00:04:01.949
So what happened with that tiger?

00:04:01.949 --> 00:04:03.090
- [Voiceover] That's a long story.

00:04:03.090 --> 00:04:04.895
- [Voiceover] All right,
we'll talk about that later.

00:04:04.895 --> 00:04:06.038
You can learn anything.

00:04:06.038 --> 00:04:06.871
Dave out.

00:04:06.871 --> 00:04:08.519
- [Voiceover] Paige out.


How to use italics and underlines

WEBVTT
Kind: captions
Language: en

00:00:00.186 --> 00:00:01.019
- [Voiceover] Hello, grammarians.

00:00:01.019 --> 00:00:01.852
Hello, Paige.

00:00:01.852 --> 00:00:02.685
- [Voiceover] Hi, David.

00:00:02.685 --> 00:00:04.567
- [Voiceover] So, Paige,
have you ever heard

00:00:04.567 --> 00:00:06.817
of this man Aldus Manutius?

00:00:07.861 --> 00:00:08.955
- [Voiceover] I don't think I have.

00:00:08.955 --> 00:00:09.872
That's a pretty cool name, though.

00:00:09.872 --> 00:00:12.379
- [Voiceover] His given name
was actually Aldo Manuzio.

00:00:12.379 --> 00:00:15.662
He was a Venetian printer around 1500.

00:00:15.662 --> 00:00:19.385
And this guy invented the italic typeface.

00:00:19.385 --> 00:00:20.218
- [Voiceover] What?

00:00:20.218 --> 00:00:21.051
- [Voiceover] Yeah.

00:00:21.051 --> 00:00:24.760
So, italic is this word that
really just comes from Italy,

00:00:24.760 --> 00:00:28.092
right, so he's from Venice,
which is in modern day Italy.

00:00:28.092 --> 00:00:32.471
And what it refers to is text
that is kind of on its side,

00:00:32.471 --> 00:00:33.554
so like this.

00:00:34.573 --> 00:00:36.295
Kind of slopes to the right.

00:00:36.295 --> 00:00:37.565
That's italic.

00:00:37.565 --> 00:00:39.826
And he was one of the first printers,

00:00:39.826 --> 00:00:42.972
so he had this movable type
printing machine, this press.

00:00:42.972 --> 00:00:45.298
And he developed these letters.

00:00:45.298 --> 00:00:47.888
He cut them into pieces of metal

00:00:47.888 --> 00:00:51.502
and he developed this
kind of slanted style.

00:00:51.502 --> 00:00:54.498
And today we call this
the italic typeface.

00:00:54.498 --> 00:00:55.331
- [Voiceover] Wow!

00:00:55.331 --> 00:00:56.164
- [Voiceover] Actually, technically,

00:00:56.164 --> 00:00:58.513
we'd call this the italic script,

00:00:58.513 --> 00:01:02.670
because it's based on
an earlier kind of hand

00:01:02.670 --> 00:01:05.074
from calligraphy, but all the same,

00:01:05.074 --> 00:01:09.324
today we have this thing
in text called italics.

00:01:09.324 --> 00:01:11.501
So, okay, so we've got our typeface here.

00:01:11.501 --> 00:01:13.098
And it comes in two flavors.

00:01:13.098 --> 00:01:14.886
We've got Roman, which is
kind of standing right up,

00:01:14.886 --> 00:01:17.391
and italic, which is slanted to the right.

00:01:17.391 --> 00:01:20.457
Paige, how do we use italics in English?

00:01:20.457 --> 00:01:23.741
- [Voiceover] Italics
have two uses in English,

00:01:23.741 --> 00:01:26.601
one is to emphasize and
the other is to indicate

00:01:26.601 --> 00:01:28.895
that something is a title.

00:01:28.895 --> 00:01:30.771
- [Voiceover] So we use it
for emphasis and titling.

00:01:30.771 --> 00:01:32.933
Let's dive into the first one.

00:01:32.933 --> 00:01:34.200
- [Voiceover] Okay.

00:01:34.200 --> 00:01:36.198
- [Voiceover] So if we're
trying to use italics

00:01:36.198 --> 00:01:38.781
to emphasize something in text,

00:01:40.145 --> 00:01:42.628
we would just use it to set off one word.

00:01:42.628 --> 00:01:43.961
So, for example,

00:01:45.122 --> 00:01:47.866
That's a really spicy pepper.

00:01:47.866 --> 00:01:50.893
- [Voiceover] Yeah, the word
really is said very strongly

00:01:50.893 --> 00:01:51.726
in that sentence.

00:01:51.726 --> 00:01:53.578
- [Voiceover] Mm-hmm.

00:01:53.578 --> 00:01:56.321
Okay, and use case number two for titling.

00:01:56.321 --> 00:01:57.154
And you know what?

00:01:57.154 --> 00:02:01.096
To make it easier on ourselves,
let's pull up a text box.

00:02:01.096 --> 00:02:03.136
- [Voiceover] Yeah, it's kind
of hard to write italics.

00:02:03.136 --> 00:02:05.112
- [Voiceover] Yeah, which
is actually gonna lead us

00:02:05.112 --> 00:02:08.152
into another part of this video.

00:02:08.152 --> 00:02:09.663
You'll see, but right now, okay,

00:02:09.663 --> 00:02:12.815
I got my italic type selected.

00:02:12.815 --> 00:02:14.735
So how do titles work?

00:02:14.735 --> 00:02:16.285
- [Voiceover] So if you
got a title of something,

00:02:16.285 --> 00:02:18.035
like book or a movie.

00:02:19.485 --> 00:02:21.821
- [Voiceover] So, okay, so
we got a book, The Hobbit,

00:02:21.821 --> 00:02:23.071
we got a movie,

00:02:26.617 --> 00:02:30.556
and we have a full-length musical work

00:02:30.556 --> 00:02:31.908
or like a symphony or an album

00:02:31.908 --> 00:02:35.353
like Kate Bush's Hounds of Love.

00:02:35.353 --> 00:02:39.520
All of those things can be
rendered in italic type.

00:02:43.269 --> 00:02:47.269
So, okay, so for full
works, then, so for books,

00:02:48.823 --> 00:02:49.656
movies,

00:02:51.205 --> 00:02:55.809
and full-length musical works
like an album or a symphony.

00:02:55.809 --> 00:02:59.007
- [Voiceover] Okay, so like
a song or a chapter of a book

00:02:59.007 --> 00:03:01.328
is not considered a full work.

00:03:01.328 --> 00:03:02.161
- [Voiceover] Correct.

00:03:02.161 --> 00:03:04.202
- [Voiceover] Okay.

00:03:04.202 --> 00:03:06.079
- [Voiceover] Now, as
you pointed out, Paige,

00:03:06.079 --> 00:03:08.749
like the difference between
spicy and really here,

00:03:08.749 --> 00:03:11.494
not the easiest to discern.

00:03:11.494 --> 00:03:13.414
So if I were going to be writing
out the title of something

00:03:13.414 --> 00:03:16.805
just by hand, it's pretty
hard to do italics.

00:03:16.805 --> 00:03:18.852
So what we do when we're
writing things by hand

00:03:18.852 --> 00:03:20.852
is we use the underline.

00:03:21.976 --> 00:03:22.809
- [Voiceover] Yeah.

00:03:22.809 --> 00:03:25.674
So the underline means
the same thing as italics.

00:03:25.674 --> 00:03:28.820
You use it for titles, for full works,

00:03:28.820 --> 00:03:31.745
but it's a lot easier to
hand write and underline

00:03:31.745 --> 00:03:34.084
than it is to hand write italics.

00:03:34.084 --> 00:03:35.337
- [Voiceover] Paige,
what's your favorite book?

00:03:35.337 --> 00:03:38.606
- [Voiceover] Let's say The Giver.

00:03:38.606 --> 00:03:39.689
That's a great book.

00:03:39.689 --> 00:03:41.316
- [Voiceover] That is a great book.

00:03:41.316 --> 00:03:42.588
All right, so I'm writing The Giver.

00:03:42.588 --> 00:03:44.479
I want to indicate that it's a full title.

00:03:44.479 --> 00:03:46.823
Voop, put that underline underneath it.

00:03:46.823 --> 00:03:48.406
The Give is Paige's

00:03:52.089 --> 00:03:53.256
favorite book.

00:03:57.842 --> 00:04:00.787
So, the way we're underlining The Giver

00:04:00.787 --> 00:04:02.600
is the same functionally

00:04:02.600 --> 00:04:04.809
as the way we've italicized The Hobbit,

00:04:04.809 --> 00:04:07.781
The Thief and the Cobbler,
or Hounds of Love.

00:04:07.781 --> 00:04:08.614
- [Voiceover] That's right.

00:04:08.614 --> 00:04:10.728
- [Voiceover] So, that's
where the italic typeface

00:04:10.728 --> 00:04:12.788
comes from is this guy Aldus Manutius.

00:04:12.788 --> 00:04:14.201
This is how we use it today.

00:04:14.201 --> 00:04:15.188
You can learn anything.

00:04:15.188 --> 00:04:16.109
David out.

00:04:16.109 --> 00:04:17.111
- [Voiceover] Paige out.