Question

question given the function y = x^5 - 1/2 x^(-2) + 1/4, find d^2y/dx^2 in simplified form.

Explanation:

Step1: Find the first - derivative

Use the power rule $\frac{d}{dx}(x^n)=nx^{n - 1}$.
$y=x^{5}-\frac{1}{2}x^{-2}+\frac{1}{4}$
$\frac{dy}{dx}=5x^{4}- \frac{1}{2}\times(-2)x^{-3}+0=5x^{4}+x^{-3}$

Step2: Find the second - derivative

Differentiate $\frac{dy}{dx}$ with respect to $x$ again using the power rule.
$\frac{d^{2}y}{dx^{2}}=5\times4x^{3}+(-3)x^{-4}=20x^{3}-3x^{-4}$
$=\ 20x^{3}-\frac{3}{x^{4}}$

Answer:

$20x^{3}-\frac{3}{x^{4}}$