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(3t + 2).

Example 2

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

Example 3

Find the derivative of y = sec43x.

Continues below

Exercises

1. Find the derivative of y = csc2(2x2).

2. Find the derivative of y = sec2 2x.

3. Find the derivative of 3 cot(x + y) = cos y2.