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

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) )

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) ).