Question

if $f(x)=5 + \frac{5}{x}+\frac{6}{x^{2}}$, find $f(x)$.

Explanation:

Step1: Rewrite the function

Rewrite $f(x)=5 + \frac{5}{x}+\frac{6}{x^{2}}$ as $f(x)=5 + 5x^{-1}+6x^{-2}$.

Step2: Apply the power - rule for differentiation

The power - rule states that if $y = ax^{n}$, then $y^\prime=anx^{n - 1}$. For the constant term $5$, its derivative is $0$ (since the derivative of a constant $C$ is $0$). For the term $5x^{-1}$, its derivative is $5\times(-1)x^{-1 - 1}=-5x^{-2}$. For the term $6x^{-2}$, its derivative is $6\times(-2)x^{-2 - 1}=-12x^{-3}$.

Step3: Combine the derivatives

$f^\prime(x)=0-5x^{-2}-12x^{-3}=-\frac{5}{x^{2}}-\frac{12}{x^{3}}$.

Answer:

$-\frac{5}{x^{2}}-\frac{12}{x^{3}}$