# Derivative of sec x

How to find the derivative of the given function by using the derivative of sec x: formula, 1 example, and its solution.

## Formula

The derivative of sec x, [sec x]',

is sec x tan x.

## Exampley = sec^{3} x, y' = ?

Solution

Solution (Detail)

y = sec^{3} x means y = (sec x)^{3}.

This is a composite function of

y = (whole)^{3} and (whole) = sec x.

So the derivative of the composite function is,

the derivative of the outer function (sec x)^{3}, 3(sec x)^{2}

times,

the derivative of the inner function sec x, sec x tan x.

(sec x)^{2}⋅(sec x) = (sec x)^{3} = sec^{3} x

So the right side is 3 sec^{3} x tan x.

So

3 sec^{3} x tan x

is the derivative of sec^{3} x.