Question

find the derivative of the following function. g(x) = \frac{x^{5}+5x^{2}+7}{x^{5}} g(x) = \square

Explanation:

Step1: Simplify the function

First, rewrite $g(x)=\frac{x^{5}+5x^{2}+7}{x^{5}}$ as $g(x)=1 + 5x^{-3}+7x^{-5}$ using the rule $\frac{a + b + c}{d}=\frac{a}{d}+\frac{b}{d}+\frac{c}{d}$.

Step2: Apply the power - rule for derivatives

The power - rule states that if $y = x^{n}$, then $y^\prime=nx^{n - 1}$.
For the first term, the derivative of a constant 1 is 0.
For the second term, if $y = 5x^{-3}$, then $y^\prime=5\times(-3)x^{-3 - 1}=-15x^{-4}$.
For the third term, if $y = 7x^{-5}$, then $y^\prime=7\times(-5)x^{-5 - 1}=-35x^{-6}$.

Step3: Combine the derivatives

$g^\prime(x)=0-15x^{-4}-35x^{-6}=-\frac{15}{x^{4}}-\frac{35}{x^{6}}$

Answer:

$-\frac{15}{x^{4}}-\frac{35}{x^{6}}$