Question

given the function $f(x)=x^{-2}-\frac{1}{4}x^{5}-\frac{1}{2}$, find $f(x)$ in simplified form.

Explanation:

Step1: Find the first - derivative

Use the power rule $\frac{d}{dx}(x^n)=nx^{n - 1}$.
$f'(x)=\frac{d}{dx}(x^{-2}-\frac{1}{4}x^{5}-\frac{1}{2})=-2x^{-3}-\frac{5}{4}x^{4}$

Step2: Find the second - derivative

Differentiate $f'(x)$ again using the power rule.
$f''(x)=\frac{d}{dx}(-2x^{-3}-\frac{5}{4}x^{4})=6x^{-4}-5x^{3}=\frac{6}{x^{4}}-5x^{3}$

Answer:

$\frac{6}{x^{4}}-5x^{3}$