Derivative of Arcsine Function/Corollary

Corollary to Derivative of Arcsine Function

Let $a \in \R$ be a constant

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

Let $\arcsin \left({\dfrac x a}\right)$ be the arcsine of $\dfrac x a$.

Then:

$\dfrac {\mathrm d \left({\arcsin \left({\frac x a}\right)}\right)} {\mathrm d x} = \dfrac 1 {\sqrt {a^2 - x^2}}$

Proof

 $\displaystyle \frac {\mathrm d \left({\arcsin \left({\frac x a}\right)}\right)} {\mathrm d x}$ $=$ $\displaystyle \frac 1 a \frac 1 {\sqrt {1 - \left({\frac x a}\right)^2} }$ Derivative of Arcsine Function and Derivative of Function of Constant Multiple $\displaystyle$ $=$ $\displaystyle \frac 1 a \frac 1 {\sqrt {\frac {a^2 - x^2} {a^2} } }$ $\displaystyle$ $=$ $\displaystyle \frac 1 a \frac a {\sqrt {a^2 - x^2} }$ $\displaystyle$ $=$ $\displaystyle \frac 1 {\sqrt {a^2 - x^2} }$

$\blacksquare$