Question

given the function f(x)=-\frac{1}{4}x^{-4}-\frac{3}{10}x-\frac{5}{3}+x^{2}, find f(x) in simplified form.

Explanation:

Step1: Recall power - rule for differentiation

The power - rule states that if $y = ax^n$, then $y^\prime=anx^{n - 1}$.

Step2: Find the first - derivative $f^\prime(x)$

For $y =-\frac{1}{4}x^{-4}$, $y^\prime=-\frac{1}{4}\times(-4)x^{-4 - 1}=x^{-5}$; for $y =-\frac{3}{10}x$, $y^\prime=-\frac{3}{10}$; for $y =-\frac{5}{3}$ (a constant), $y^\prime = 0$; for $y=x^{2}$, $y^\prime = 2x$. So, $f^\prime(x)=x^{-5}-\frac{3}{10}+2x$.

Step3: Find the second - derivative $f^{\prime\prime}(x)$

For $y = x^{-5}$, $y^\prime=-5x^{-5 - 1}=-5x^{-6}$; for $y =-\frac{3}{10}$ (a constant), $y^\prime = 0$; for $y = 2x$, $y^\prime=2$. So, $f^{\prime\prime}(x)=-5x^{-6}+2=\frac{-5}{x^{6}}+2$.

Answer:

$f^{\prime\prime}(x)=\frac{-5}{x^{6}}+2$