Question

if ( f(x)=sin(x^{3}) ), find ( f^{prime}(x) ).
( f^{prime}(x)= )

Explanation:

Step1: Apply the chain rule

Let \(u = x^{3}\), then \(f(x)=\sin(u)\). The chain rule states that \(\frac{df}{dx}=\frac{df}{du}\cdot\frac{du}{dx}\).
For \(y = \sin(u)\), \(\frac{dy}{du}=\cos(u)\).
For \(u=x^{3}\), \(\frac{du}{dx}=3x^{2}\).

Step2: Substitute back

Substitute \(u = x^{3}\) into \(\frac{df}{du}\cdot\frac{du}{dx}\). We get \(f^{\prime}(x)=\cos(x^{3})\cdot3x^{2}\).

Answer:

\(3x^{2}\cos(x^{3})\)