Derivative of Composite Function/Examples/Root of sin x

Example of Derivative of Composite Function

$\map {\dfrac \d {\d x} } {\sqrt {\sin x} } = \dfrac {\cos x} {2 \sqrt {\sin x} }$

Proof

Let $u = \sin x$.

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

Thus by definition of square root we have:

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

and so:

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

\(=\)

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

Derivative of Composite Function

\(\ds \)

\(=\)

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

Power Rule for Derivatives, Derivative of Sine Function

\(\ds \)

\(=\)

\(\ds \dfrac {\cos x} {2 \sqrt {\sin x} }\)

simplification

$\blacksquare$

Sources

• 1953: L. Harwood Clarke: A Note Book in Pure Mathematics ... (previous) ... (next): $\text {II}$. Calculus: Differentiation: Exercises $\text {IX}$: $2$.