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## Finding the derivatives of tangent, cotangent, secant, and/or cosecant functions

# Derivatives of tan(x) and cot(x)

AP.CALC:

FUN‑3 (EU)

, FUN‑3.B (LO)

, FUN‑3.B.3 (EK)

## Video transcript

- [Voiceover] We already
know the derivatives of sine and cosine. We know that the derivative
with respect to x of sine of x is equal to cosine of x. We know that the derivative
with respect to x of cosine of x is equal
to negative sine of x. And so what we want to do in this video is find the derivatives of the
other basic trig functions. So, in particular, we
know, let's figure out what the derivative with respect to x, let's first do tangent of x. Tangent of x, well this is the same thing as trying to find the
derivative with respect to x of, well, tangent of x is just sine of x, sine of x over cosine of x. And since it can be expressed as the quotient of two functions, 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 the derivative of the top function, 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, so I can put the sine of x there, but where the negative
can just cancel that out. And it's going to be over, over the bottom function squared. So cosine squared of x. Now, what is this? Well, what we have here, this
is just a cosine squared of x, this is just sine squared of x. And we know from the Pythagorean identity, and this is really just out of, comes out of the unit circle definition, the cosine squared of x
plus sine squared of x, well that's gonna be
equal to one for any x. So all of this is equal to one. And so we end up with one
over cosine squared x, which is the same thing as,
which is the same thing as, the secant of x squared. One over cosine of x is secant, so this is just secant of x squared. So that was pretty straightforward. Now, let's just do the inverse of the, or you could say the
reciprocal, I should say, of the tangent function,
which is the cotangent. Oh, that was fun, so let's do that, d dx of cotangent, not cosine, of cotangent of x. 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, 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. That is negative sine of x times the bottom function, so times sine of x, sine of x, minus, minus the top function, cosine of x, 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. So sine of x squared. Now what does this simplify to? Up here, let's see, this
is sine squared of x, we have a negative there, minus cosine squared of x. But we could factor out the negative and this would be
negative sine squared of x plus cosine squared of x. Well, this is just one by
the Pythagorean identity, and so this is negative
one over sine squared x, negative one over sine squared x. And that is the same thing as negative cosecant squared, I'm running out of space, of x. There you go.

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