Name: Derivatives of tan(x) and cot(x) | Derivative rules | AP Calculus AB | Khan Academy
Uploaded: 2022-07-12T12:41:28.000Z
Duration: 4 min 38 s
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- [Voiceover] We already know the derivatives

The subjunctive

We know that the derivative with respect to x

of cosine of x is equal to negative sine of x.

is find the derivatives of the other basic trig functions.

So, in particular, we know, let's figure out

Tangent of x, well this is the same thing

as trying to find the derivative with respect to x of,

we can apply the quotient rule here to evaluate this,

or to figure out what this is going to be.

The quotient rule tells us that this is going to be

which we know is cosine of x times the bottom function

which is cosine of x, so times cosine of x

minus, minus the top function, which is sine of x,

sine of x, times the derivative of the bottom function.

So the derivative of cosine of x is negative sine of x,

but where the negative can just cancel that out.

Well, what we have here, this is just a cosine squared of x,

And we know from the Pythagorean identity,

the cosine squared of x plus sine squared of x,

well that's gonna be equal to one for any x.

And so we end up with one over cosine squared x,

which is the same thing as, which is the same thing as,

or you could say the reciprocal, I should say,

of the tangent function, which is the cotangent.

Well, same idea, that's the derivative with respect to x,

and this time, let me make some sufficiently large brackets.

So now this is cosine of x over sine of x,

But once again, we can use the quotient rule here,

so this is going to be the derivative of the top function

which is negative, use that magenta color.

cosine of x, times the derivative of the bottom function

which is just going to be another cosine of x,

and then all of that over the bottom function squared.

Up here, let's see, this is sine squared of x,

and this would be negative sine squared of x

Well, this is just one by the Pythagorean identity,

and so this is negative one over sine squared x,

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Derivatives of tan(x) and cot(x) | Derivative rules | AP Calculus AB | Khan Academy

figure

negative

straightforward

express