Question

let ( f(x)=ln(x^{4})) ( f(x)=4) ( f(e^{4}) = 0) 2 of the answers are not correct.

Explanation:

Step1: Differentiate $f(x)$

First, use the chain - rule. If $y = \ln(u)$ and $u=x^{4}$, then $\frac{dy}{dx}=\frac{1}{u}\cdot\frac{du}{dx}$. Since $u = x^{4}$, $\frac{du}{dx}=4x^{3}$, so $f^{\prime}(x)=\frac{4x^{3}}{x^{4}}=\frac{4}{x}$.

Step2: Evaluate $f^{\prime}(e^{4})$

Substitute $x = e^{4}$ into $f^{\prime}(x)$. We get $f^{\prime}(e^{4})=\frac{4}{e^{4}}$.

Answer:

$f^{\prime}(x)=\frac{4}{x}$, $f^{\prime}(e^{4})=\frac{4}{e^{4}}$