Question

find ( f^{prime}(x) ).

( f(x)=left(5 x^{3}+4
ight)^{5} )

( f^{prime}(x)= )

Explanation:

Step1: Apply the chain rule

Let \( u = 5x^{3}+4 \), then \( f(x)=u^{5} \). The chain rule states \( f^{\prime}(x)=\frac{df}{du}\cdot\frac{du}{dx} \).
First, find \( \frac{df}{du} \): \( \frac{df}{du}=\frac{d}{du}(u^{5}) = 5u^{4} \).

Step2: Find \( \frac{du}{dx} \)

\( \frac{du}{dx}=\frac{d}{dx}(5x^{3}+4)=15x^{2} \).

Step3: Substitute \( u \) back

Substitute \( u = 5x^{3}+4 \) into \( \frac{df}{du} \), then \( f^{\prime}(x)=5(5x^{3}+4)^{4}\cdot15x^{2} \).
Simplify the expression: \( f^{\prime}(x)=75x^{2}(5x^{3}+4)^{4} \).

Answer:

\( 75x^{2}(5x^{3}+4)^{4} \)