# Derivative of csc x

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

## Formula

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

is -csc x cot x.

## Exampley = csc (1 - x^{2}), y' = ?

Solution

Solution (Detail)

y = csc (1 - x^{2})

is a composite function of

y = csc (whole) and (whole) = 1 - x^{2}.

So the derivative of the composite function is,

the derivative of the outer function csc (1 - x^{2}), -csc (1 - x^{2}) cot (1 - x^{2})

times,

the derivative of the inner function 1 - x^{2}, -2x^{1}.

-(-2x^{1}) = 2x

So the right side is

2x csc (1 - x^{2}) cot (1 - x^{2}).

So

2x csc (1 - x^{2}) cot (1 - x^{2})

is the derivative of csc (1 - x^{2}).