# 2. Derivatives of Csc, Sec and Cot Functions

by M. Bourne

By using the quotient rule and trigonometric identities, we can obtain the following derivatives:

`(d(csc x))/(dx)=-csc x cot x`

`(d(sec x))/(dx)=sec x tan x`

`(d(cot x))/(dx)=-csc^2 x`

In words, we would say:

The derivative of `csc x` is `-csc x cot x`,

The derivative of `sec x` is `sec x tan x` and

The derivative of `cot x` is `-csc^2 x`.

Explore animations of these functions with their derivatives here:

Differentiation Interactive Applet - trigonometric functions.

If *u* = *f*(*x*) is a function of *x*, then
by using the chain rule, we have:

`(d(csc u))/(dx)=-csc u\ cot u(du)/(dx)`

`(d(sec u))/(dx)=sec u\ tan u(du)/(dx)`

`(d(cot u))/(dx)=-csc^2u(du)/(dx)`

### Example 1

Find the derivative of *s* = sec(3*t* + 2).

### Example 2

Find the derivative of `x = θ^3 csc 2θ`.

### Example 3

Find the derivative
of *y* = sec^{4}3*x*.

## Exercises

1. Find the derivative of *y* = csc^{2}(2*x*^{2}).

2. Find the derivative of *y* = sec^{2} 2*x*.

3. Find the derivative of 3 cot(*x* + *y*) = cos* y*^{2}.

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