Question

derivatives of logarithmic functions: problem 1 (1 point)
let
$f(x)=ln(x^{5})$
$f^{prime}(x)=\frac{5x^{4}}{x^{5}}$
$f^{prime}(e^{2})=$

Explanation:

Step1: Simplify the derivative

We have $f^{\prime}(x)=\frac{5x^{4}}{x^{5}}$. Simplify the right - hand side by using the rule $\frac{x^{m}}{x^{n}}=x^{m - n}$. So $\frac{5x^{4}}{x^{5}}=5x^{4-5}= \frac{5}{x}$.

Step2: Substitute $x = e^{2}$

Substitute $x = e^{2}$ into $f^{\prime}(x)=\frac{5}{x}$. We get $f^{\prime}(e^{2})=\frac{5}{e^{2}}$.

Answer:

$\frac{5}{e^{2}}$