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## Derivatives of tan(x), cot(x), sec(x), and csc(x)

# Derivatives of sec(x) and csc(x)

AP.CALC:

FUN‑3 (EU)

, FUN‑3.B (LO)

, FUN‑3.B.3 (EK)

## Video transcript

- [Voiceover] In a previous
video we used the quotient rule in order to find the derivatives of tangent of x and cotangnet of x. And what I what to do in
this video is to keep going and find the derivatives of
secant of x and cosecant of x. So let's start with secant of x. The derivative with respect
to x of secant of x. Well, secant of x is the same thing as so we're going to find the
derivative with respect to x of secant of x is the same thing as one over, one over the cosine of x. And that's just the definition of secant. And there's multiple
ways you could do this. When you learn the chain rule, that actually might be a
more natural thing to use to evaluate the derivative here. But we know the quotient rule, so we will apply the quotient rule here. And it's no coincidence that
you get to the same answer. The quotient rule actually can be derived based on the chain rule
and the product rule. But I won't keep going into that. Let's just apply the quotient
rule right over here. So this derivative is
going to be equal to, it's going to be equal to
the derivative of the top. Well, what's the derivative
of one with respect to x? Well, that's just zero. Times the function on the bottom. So, times cosine of x. Cosine of x. Minus, minus the function on the top. Well, that's just one. Times the derivative on the bottom. Well, the derivative on the bottom is, the derivative of cosine
of x is negative sine of x. So we could put the sine of x there. But it's negative sine of x, so you have a minus and
it'll be a negative, so we can just make that a positive. And then all of that over the function on the bottom squared. So, cosine of x, squared. And so zero times cosine of x, that is just zero. And so all we are left with is sine of x over cosine of x squared. And there's multiple ways
that you could rewrite this if you like. You could say that this
is same thing as sine of x over cosine of x times
one over cosine of x. And of course this is tangent of x, times secant of x. Secant of x. So you could say derivative
of secant of x is sine of x over cosine-squared of x. Or it is tangent of x
times the secant of x. So now let's do cosecant. So the derivative with
respect to x of cosecant of x. Well, that's the same
thing as the derivative with respect to x of one over sine of x. Cosecant is one over sine of x. I remember that because
you think it's cosecant. Maybe it's the reciprocal
of cosine, but it's not. It's the opposite of
what you would expect. Cosine's reciprocal isn't
cosecant, it is secant. Once again, opposite of
what you would expect. That starts with an s,
this starts with a c. That starts with a c,
that starts with an s. It's just way it happened to be defined. But anyway, let's just evaluate this. Once again, we'll do the quotient rule, but you could also do
this using the chain rule. So it's going to be the derivative of the expression
on top, which is zero, times the expression on the
bottom, which is sine of x. Sine of x. Minus the expression on
top, which is just one. Times the derivative of the
expression on the bottom, which is cosine of x. All of that over the expression
on the bottom squared. Sine-squared of x. That's zero. So we get negative cosine of x over sine-squared of x. So that's one way to think about it. Or if you like, you could do this, the same thing we did over here, this is the same thing
as negative cosine of x over sine of x, times one over sine of x. And this is negative cotangent of x. Negative cotangent of x, times, maybe I'll write it this way, times one over sine of x is cosecant of x. Cosecant of x. So, which ever one you find more useful.