# Derivative of Arctangent Function/Corollary

## Corollary to Derivative of Arctangent Function

Let $x \in \R$.

Let $\map \arctan {\dfrac x a}$ denote the arctangent of $\dfrac x a$.

Then:

$\dfrac {\map \d {\map \arctan {\frac x a} } } {\d x} = \dfrac a {a^2 + x^2}$

## Proof

 $\ds \frac {\map \d {\map \arctan {\frac x a} } } {\d x}$ $=$ $\ds \frac 1 a \frac 1 {1 + \paren {\frac x a}^2}$ Derivative of Arctangent Function and Derivative of Function of Constant Multiple $\ds$ $=$ $\ds \frac 1 a \frac 1 {\frac {a^2 + x^2} {a^2} }$ $\ds$ $=$ $\ds \frac 1 a \frac {a^2} {a^2 + x^2}$ $\ds$ $=$ $\ds \frac a {a^2 + x^2}$

$\blacksquare$

## Also defined as

This result can also be reported as:

$\dfrac {\map \d {\map \arctan {\frac x a} } } {\d x} = \dfrac a {x^2 + a^2}$