Question

Use the definition of the derivative to find $f$for each function $f$in Exercises 39-54

$f\left(x\right)=\frac{x^3}{x+1}$

Short answer

The derivative is$f^{\text{'}}\left(x\right)=\frac{2x^3+3x^2}{(x+1{)}^2}$

Step-by-step solution

Step 1.Given information

Given function$f\left(x\right)=\frac{x^3}{x+1}$

Use the chain rule and calculate

Calculating, we get

$$\begin{gathered} f\left(x\right)=\frac{x^3}{x+1} \\ f\text{'}\left(x\right)=\frac{(x+1).3x^2-x^3.1}{{(x+1)}^2} \\ f\text{'}\left(x\right)=\frac{3x^3+3x^2-x^3}{{(x+1)}^2} \\ f\text{'}\left(x\right)=\frac{2x^3+3x^2}{{(x+1)}^2} \end{gathered}$$