Derivative of Arcsine of Function

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Theorem

Let $u$ be a differentiable real function of $x$ such that $\size {\map u x} < 1$, that is, $\size {\arcsin u} < \dfrac \pi 2$.

Then:

$\map {\dfrac \d {\d x} } {\arcsin u} = \dfrac 1 {\sqrt {1 - u^2} } \dfrac {\d u} {\d x}$

where $\arcsin$ denotes the real arcsine of $x$.


Proof

\(\ds \map {\frac \d {\d x} } {\arcsin u}\) \(=\) \(\ds \map {\frac \d {\d u} } {\arcsin u} \frac {\d u} {\d x}\) Chain Rule for Derivatives
\(\ds \) \(=\) \(\ds \dfrac 1 {\sqrt {1 - u^2} } \frac {\d u} {\d x}\) Derivative of Arcsine Function

$\blacksquare$


Also see


Sources

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