Question

find the derivative of the following function. g(x)=\frac{x^{5}+9x^{4}+5}{x^{5}}

Explanation:

Step1: Simplify the function

First, split the fraction: $g(x)=\frac{x^{5}}{x^{5}}+\frac{9x^{4}}{x^{5}}+\frac{5}{x^{5}} = 1 + 9x^{-1}+5x^{-5}$

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 $9x^{-1}$, its derivative is $9\times(-1)x^{-1 - 1}=-9x^{-2}$. For $5x^{-5}$, its derivative is $5\times(-5)x^{-5 - 1}=-25x^{-6}$.

Step3: Combine the derivatives

$g^\prime(x)=0-9x^{-2}-25x^{-6}=- \frac{9}{x^{2}}-\frac{25}{x^{6}}$

Answer:

$g^\prime(x)=-\frac{9}{x^{2}}-\frac{25}{x^{6}}$