Derivative of Composite Function/Examples/a^sin x

Example of Derivative of Composite Function

$\map {\dfrac \d {\d x} } {a^{\sin x} } = \cos x a^{\sin x} \ln a$

Proof

Let $u = \sin x$.

Let $y = a^u$.

Thus we have:

$y = a^{\sin x}$

and so:

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

\(=\)

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

Derivative of Composite Function

\(\ds \)

\(=\)

\(\ds a^u \ln a \cdot \cos x\)

Derivative of General Exponential Function, Derivative of Sine Function

\(\ds \)

\(=\)

\(\ds \cos x a^{\sin x} \ln a\)

simplification

$\blacksquare$

Sources

• 1953: L. Harwood Clarke: A Note Book in Pure Mathematics ... (previous) ... (next): $\text {II}$. Calculus: Differentiation: Exercises $\text {IX}$: $30$.