Question

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

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

( f^{prime}(x)=square )

Explanation:

Step1: Let \( u = 4x^{5}+2 \)

Then \( f(x)=u^{3} \)

Step2: Differentiate \( u \) and \( f(u) \)

Differentiate \( u \): \( u^\prime=\frac{d}{dx}(4x^{5}+2)=20x^{4} \)
Differentiate \( f(u) \): \( f^\prime(u)=\frac{d}{du}(u^{3}) = 3u^{2} \)

Step3: Apply the chain rule \( f^\prime(x)=f^\prime(u)\cdot u^\prime \)

Substitute \( u = 4x^{5}+2 \) and \( u^\prime=20x^{4} \), \( f^\prime(u) = 3u^{2} \) into the chain - rule formula.
\( f^\prime(x)=3(4x^{5}+2)^{2}\cdot20x^{4} \)

Step4: Simplify the expression

\( f^\prime(x)=60x^{4}(4x^{5}+2)^{2} \)

Answer:

\(60x^{4}(4x^{5}+2)^{2}\)