# Derivative of cot *x*

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

## Formula

The derivative of cot *x* is -csc^{2} *x*.

## Proof

cot *x* = 1/(tan *x*)

Trigonometric ratio - cotangent

Reciprocal rule in differentiation

Derivative of tan *x*

sec^{2} *x* = 1/(cos^{2} *x*)

Trigonometric ratio - secant

And 1/(tan^{2} *x*) = (cos^{2} *x*)/(sin^{2} *x*).

Trigonometric ratio - tangent

Cancel cos^{2} *x*.

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

Trigonometric ratio - cosecant

So -1/(sin^{2} *x*) = -csc^{2} *x*.

So [cot *x*]' = -csc^{2} *x*.

## Example

*y* = (7*x*)(tan *x*)

So *y*' is equal to,

the derivative of 7*x*, 7

times cot *x*

plus 7*x*,

times, the derivative of cot *x*, -csc^{2} *x*.

Product rule in differentiation

Power rule in differentiation (Part 1)