# Derivative of csc *x*

How to solve the derivative of csc x problems: formula, proof, example, and its solution.

## Formula

The derivative of csc *x* is -csc *x* cot *x*.

## Proof

csc *x* = 1/(sin *x*)

Trigonometric ratio - cosecant

Reciprocal rule in differentiation

Derivative of sin *x*

1/(sin *x*) = csc *x*

Trigonometric ratio - cosecant

And (cos *x*)/(sin *x*) = cot *x*.

Trigonometric ratio - cotangent

So -1/(sin *x*) ⋅ (cos *x*)/(sin *x*)

= -csc *x* cot *x*.

So [csc *x*]' = -csc *x* cot *x*.

## Example

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

So *y*' is equal to,

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

times, the derivative of the inner part, (-2*x*).

Chain rule in differentiation

Derivatives of polynomials