Derivative of Composite Function/Examples/Sine of 2 x Cubed
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Example of Derivative of Composite Function
- $\map {\dfrac \d {\d x} } {\sin^3 2 x} = 6 \sin^2 2 x \cos 2 x$
Proof
Let $u = \sin 2 x$.
Let $y = u^3$.
Thus we have:
- $y = \sin^3 2 x$
and so:
\(\ds \dfrac {\d y} {\d x}\) | \(=\) | \(\ds \dfrac {\d y} {\d u} \dfrac {\d u} {\d x}\) | Derivative of Composite Function | |||||||||||
\(\ds \) | \(=\) | \(\ds 3 u^2 \cdot 2 \cos 2 x\) | Power Rule for Derivatives, Derivative of $\sin 2 x$ | |||||||||||
\(\ds \) | \(=\) | \(\ds 6 \sin^2 2 x \cos 2 x\) | simplification |
$\blacksquare$
Sources
- 1953: L. Harwood Clarke: A Note Book in Pure Mathematics ... (previous) ... (next): $\text {II}$. Calculus: Differentiation: Exercises $\text {IX}$: $8$.