Question

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

( f(x)=left(2 x^{3}+5
ight)^{4} )

( f^{prime}(x)= )

Explanation:

Step1: Apply the chain rule

Let \(u = 2x^{3}+5\), then \(y = u^{4}\). The chain rule states that \(\frac{dy}{dx}=\frac{dy}{du}\cdot\frac{du}{dx}\).
First, find \(\frac{dy}{du}\):
If \(y = u^{4}\), then \(\frac{dy}{du}=4u^{3}\).

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

If \(u = 2x^{3}+5\), then \(\frac{du}{dx}=6x^{2}\).

Step3: Substitute \(u\) and multiply

Substitute \(u = 2x^{3}+5\) into \(\frac{dy}{du}\), and then multiply \(\frac{dy}{du}\) and \(\frac{du}{dx}\):
\(f^{\prime}(x)=\frac{dy}{du}\cdot\frac{du}{dx}=4(2x^{3}+5)^{3}\cdot6x^{2}\)
Simplify the expression: \(f^{\prime}(x)=24x^{2}(2x^{3}+5)^{3}\)

Answer:

\(24x^{2}(2x^{3}+5)^{3}\)