Question

find f(x).

f(x) = \left(2x^5 + 6\
ight)^4

Explanation:

Step1: Let \(u = 2x^{5}+6\), then \(f(x)=u^{4}\)

By the chain - rule \(\frac{df}{dx}=\frac{df}{du}\cdot\frac{du}{dx}\)
For \(y = u^{4}\), \(\frac{df}{du}=4u^{3}\) (using the power rule \(\frac{d}{du}(u^{n})=nu^{n - 1}\))
For \(u = 2x^{5}+6\), \(\frac{du}{dx}=2\times5x^{4}=10x^{4}\) (using the power rule \(\frac{d}{dx}(ax^{n})=nax^{n - 1}\))

Step2: Substitute \(u\) back into the chain - rule formula

\(f^{\prime}(x)=\frac{df}{du}\cdot\frac{du}{dx}=4(2x^{5}+6)^{3}\cdot10x^{4}\)
Simplify the expression: \(f^{\prime}(x)=40x^{4}(2x^{5}+6)^{3}\)

Answer:

\(f^{\prime}(x)=40x^{4}(2x^{5}+6)^{3}\)