# Derivative of Cotangent Function

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## Contents

## Theorem

- $D_x \left({\cot x}\right) = -\csc^2 x = \dfrac {-1} {\sin^2 x}$

where $\sin x \ne 0$.

### Corollary

- $D_x \left({\cot a x}\right) = -a \csc^2 a x$

## Proof

From the definition of the cotangent function:

- $\cot x = \dfrac {\cos x} {\sin x}$

From Derivative of Sine Function:

- $D_x \left({\sin x}\right) = \cos x$

From Derivative of Cosine Function:

- $D_x \left({\cos x}\right) = -\sin x$

Then:

\(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle D_x \left({\cot x}\right)\) | \(=\) | \(\displaystyle \) | \(\) | \(\displaystyle \) | \(\displaystyle \frac {\sin x \left({-\sin x}\right) - \cos x \cos x} {\sin^2 x}\) | \(\displaystyle \) | \(\displaystyle \) | Quotient Rule for Derivatives | ||

\(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \) | \(=\) | \(\displaystyle \) | \(\) | \(\displaystyle \) | \(\displaystyle \frac {-\left({\sin^2 x + \cos^2 x}\right)} {\sin^2 x}\) | \(\displaystyle \) | \(\displaystyle \) | |||

\(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \) | \(=\) | \(\displaystyle \) | \(\) | \(\displaystyle \) | \(\displaystyle \frac {-1} {\sin^2 x}\) | \(\displaystyle \) | \(\displaystyle \) | Sum of Squares of Sine and Cosine |

This is valid only when $\sin x \ne 0$.

The result follows from the definition of the cosecant function.

$\blacksquare$

## Sources

- Murray R. Spiegel:
*Mathematical Handbook of Formulas and Tables*(1968)... (previous)... (next): $\S 13$: Derivatives of Trigonometric and Inverse Trigonometric Functions: $13.17$