Question

find the derivative of the following function.
g(x) = \frac{x^{6}+7x^{3}+2}{x^{6}}
g(x) =

Explanation:

Step1: Simplify the function

First, split the fraction: $g(x)=\frac{x^{6}}{x^{6}}+\frac{7x^{3}}{x^{6}}+\frac{2}{x^{6}} = 1 + 7x^{- 3}+2x^{-6}$.

Step2: Apply the power - rule for derivatives

The power - rule states that if $y = x^{n}$, then $y^\prime=nx^{n - 1}$.
For the constant term $1$, its derivative is $0$.
For the term $7x^{-3}$, its derivative is $7\times(-3)x^{-3 - 1}=-21x^{-4}$.
For the term $2x^{-6}$, its derivative is $2\times(-6)x^{-6 - 1}=-12x^{-7}$.

Step3: Combine the derivatives

$g^\prime(x)=0-21x^{-4}-12x^{-7}$.

Answer:

$- \frac{21}{x^{4}}-\frac{12}{x^{7}}$