Derivative of Composite Function/Examples/Root of Arcsine of x

Example of Derivative of Composite Function

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

Proof

Let $u = \arcsin x$.

Let $y = u^{1/2}$.

Thus we have:

$y = \paren {\arcsin x}^{1/2}$

and so:

\(\ds \dfrac {\d y} {\d x}\)

\(=\)

\(\ds \dfrac {\d y} {\d u} \dfrac {\d u} {\d x}\)

Derivative of Composite Function

\(\ds \)

\(=\)

\(\ds \dfrac 1 {2 \sqrt u} \cdot \dfrac 1 {\sqrt {1 - x^2} }\)

Power Rule for Derivatives, Derivative of Arcsine Function

\(\ds \)

\(=\)

\(\ds \dfrac 1 {2 \sqrt {\paren {1 - x^2} \arcsin x } }\)

simplification

$\blacksquare$

Sources

• 1953: L. Harwood Clarke: A Note Book in Pure Mathematics ... (previous) ... (next): $\text {II}$. Calculus: Differentiation: Exercises $\text {IX}$: $11$.