Question

if (f(x)=\frac{-8x^{5}+2x^{4}+7x^{3}}{x^{4}}). find (f(x)). (f(x)=)

Explanation:

Step1: Simplify the function

First, divide each term in the numerator by $x^{4}$:
$f(x)=\frac{-8x^{5}}{x^{4}}+\frac{2x^{4}}{x^{4}}+\frac{7x^{3}}{x^{4}}=-8x + 2+\frac{7}{x}=-8x + 2+7x^{-1}$

Step2: Differentiate term - by - term

The derivative of a constant is 0, the derivative of $ax^{n}$ is $nax^{n - 1}$.
The derivative of $-8x$ is $-8\times1\times x^{1 - 1}=-8$.
The derivative of the constant 2 is 0.
The derivative of $7x^{-1}$ is $7\times(-1)\times x^{-1 - 1}=-7x^{-2}=-\frac{7}{x^{2}}$.

Step3: Combine the derivatives

$f^{\prime}(x)=-8+0-\frac{7}{x^{2}}=-8-\frac{7}{x^{2}}$

Answer:

$-8-\frac{7}{x^{2}}$