# Derivative of Arccosine Function/Corollary

## Corollary to Derivative of Arccosine Function

Let $a \in \R$ be a constant

Let $x \in \R$ be a real number such that $x^2 < a^2$.

Let $\map \arccos {\dfrac x a}$ be the arccosine of $\dfrac x a$.

Then:

$\map {\dfrac \d {\d x} } {\map \arccos {\dfrac x a} } = \dfrac {-1} {\sqrt {a^2 - x^2} }$

## Proof

 $\ds \map {\dfrac \d {\d x} } {\map \arccos {\dfrac x a} }$ $=$ $\ds \frac 1 a \frac {-1} {\sqrt {1 - \paren {\frac x a}^2} }$ Derivative of Arccosine Function and Derivative of Function of Constant Multiple $\ds$ $=$ $\ds \frac 1 a \frac {-1} {\sqrt {\frac {a^2 - x^2} {a^2} } }$ $\ds$ $=$ $\ds \frac 1 a \frac {-a} {\sqrt {a^2 - x^2} }$ $\ds$ $=$ $\ds \frac {-1} {\sqrt {a^2 - x^2} }$

$\blacksquare$