Alright in this video, let's find the derivative...
of the secant of x.
How would we go about doing this?
Well sec(x) is another way of writing 1/cos(x)...
and since we have a quotient here,
the quotient rule would aptly apply...
so written in short hand, the quotient rule is:
y-dash or y-prime... I'm going to call it y-dash...
is equal to (vu' - uv') / v^2.
And in this case, we have y = sec(x), and...
I'm going to let u equal the numerator of 1
and v equal the denominator: cos(x).
So now if we apply the quotient rule...
for the first part, we have v, so cos(x) remains unchanged...
multiplied by the derivative of u.
So the derivative of 1, a constant, is zero...
then minus u, which is one, times the derivative of v...
which is the derivative of cos(x)...
and that's equal to -sin(x)...
all over v^2, which is cos^2(x).
So the result is positive sin(x)...
divided by the cos^2(x).
And I can rewrite this as...
[sin(x)/cos(x)]*1/cos(x).
And then I can realise that this part is equal to...
the tangent of x... and this part is equal to again...
the secant of x!
So the derivative of sec(x) is equal to...
tan(x)*sec(x)
which I think is a pretty cool result.
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