f $f(x)=\frac{7 x^{5}-7 x^{4}-7 x^{3}}{x^{4}}$, find $f^{prime}(x)$

Explanation:

Step1: Simplify the function

$$ \begin{align*} f(x)&=\frac{7x^{5}-7x^{4}-7x^{3}}{x^{4}}\ &=\frac{7x^{5}}{x^{4}}-\frac{7x^{4}}{x^{4}}-\frac{7x^{3}}{x^{4}}\ &=7x - 7-\frac{7}{x}\ &=7x-7 - 7x^{-1} \end{align*} $$

Step2: Differentiate term - by - term

Using the power rule $\frac{d}{dx}(x^{n})=nx^{n - 1}$:

• The derivative of $7x$ is $7\times1\times x^{1-1}=7$
• The derivative of $-7$ (a constant) is $0$
• The derivative of $-7x^{-1}$ is $-7\times(-1)x^{-1 - 1}=7x^{-2}=\frac{7}{x^{2}}$

Answer:

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