Derivative of Composite Function/Examples/Sine of 2 x Cubed

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 $\text {IX}$: $8$.