Derivative of Composite Function/Examples/a^sin x
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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 $\text {IX}$: $30$.