Question

if $f(x)=7ln(x)+\frac{6}{x}$, find $f(x)$.

Explanation:

Step1: Recall derivative rules

The derivative of $\ln(x)$ is $\frac{1}{x}$ and the derivative of $\frac{a}{x}=ax^{-1}$ is $-ax^{-2}$ (using power - rule for differentiation $\frac{d}{dx}(x^n)=nx^{n - 1}$).

Step2: Differentiate each term

The derivative of $7\ln(x)$ is $7\times\frac{1}{x}=\frac{7}{x}$ (constant multiple rule: $\frac{d}{dx}(cf(x)) = c\frac{d}{dx}(f(x))$). The derivative of $\frac{6}{x}=6x^{-1}$ is $-6x^{-2}=-\frac{6}{x^{2}}$.

Step3: Combine the derivatives

$f'(x)=\frac{7}{x}-\frac{6}{x^{2}}$.

Answer:

$\frac{7}{x}-\frac{6}{x^{2}}$