Question

replace? with an expression that will make the equation valid.
\frac{d}{dx}ln(x^{5}+9)=\frac{1}{x^{5}+9}cdot?
the missing expression is (square).

Explanation:

Step1: Apply chain - rule for differentiation

Let $u = x^{5}+9$. Then $\frac{d}{dx}\ln(u)=\frac{1}{u}\cdot\frac{du}{dx}$.

Step2: Differentiate $u = x^{5}+9$

$\frac{du}{dx}=5x^{4}$.

Step3: Substitute $u$ and $\frac{du}{dx}$ back

$\frac{d}{dx}\ln(x^{5}+9)=\frac{5x^{4}}{x^{5}+9}$.

Answer:

$\frac{5x^{4}}{x^{5}+9}$