Question

if ( f(x)=sin left(x^{4}
ight) ), find ( f^{prime}(x) ) find ( f^{prime}(3) )

Explanation:

Step1: Apply the chain rule

Let \( u = x^{4} \), then \( f(x)=\sin(u) \). The chain rule states \( \frac{df}{dx}=\frac{df}{du}\cdot\frac{du}{dx} \).
For \( f(u)=\sin(u) \), \( \frac{df}{du}=\cos(u) \); for \( u = x^{4} \), \( \frac{du}{dx}=4x^{3} \).

Step2: Substitute back

Substitute \( u = x^{4} \) into \( \frac{df}{dx}=\frac{df}{du}\cdot\frac{du}{dx} \), we get \( f^{\prime}(x)=\cos(x^{4})\cdot4x^{3}=4x^{3}\cos(x^{4}) \).

Step3: Evaluate \( f^{\prime}(3) \)

Substitute \( x = 3 \) into \( f^{\prime}(x) \). \( f^{\prime}(3)=4\times3^{3}\cos(3^{4}) \). Since \( 3^{3}=27 \) and \( 3^{4} = 81 \), then \( f^{\prime}(3)=108\cos(81) \).

Answer:

For \( f(x)=\sin(x^{4}) \), \( f^{\prime}(x) = 4x^{3}\cos(x^{4}) \) and \( f^{\prime}(3)=4\times3^{3}\cos(3^{4})=108\cos(81) \)