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:

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

$\ds \map {\dfrac \d {\d x} } {\map \arctan {\dfrac x a} }$

$=$

$\ds \frac 1 a \frac 1 {1 + \paren {\frac x a}^2}$

$\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 $

$=$

$\ds \frac a {a^2 + x^2}$

$\blacksquare$

This result can also be presented as:

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

1944: R.P. Gillespie: Integration (2nd ed.) ... (previous) ... (next): Chapter $\text {II}$: Integration of Elementary Functions: $\S 7$. Standard Integrals: $13$.