Derivative of Arccotangent Function/Corollary

Corollary to Derivative of Arccotangent Function

Let $x \in \R$.

Let $\arccot \dfrac x a$ be the arccotangent of $\dfrac x a$.

Then:

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

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

$=$

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

$\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 reported as:

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