Question

Find the derivative of $y\left(x\right)=\frac{x^3}{(x+1{)}^2}$ with respect to $x$.

• A. $\frac{2x^2}{(x+1{)}^2}-\frac{2x^3}{(x+1{)}^3}$
• B. $\frac{3x^2}{(x+1{)}^2}-\frac{2x^3}{(x+1{)}^3}$
• C. $\frac{3x^2}{(x+1{)}^2}+\frac{2x^3}{(x+1{)}^3}$
• D. $\frac{3x^2}{(x+1{)}^2}-\frac{3x^3}{(x+1{)}^3}$

Answer 1

The correct option is B $\frac{3x^2}{(x+1{)}^2}-\frac{2x^3}{(x+1{)}^3}$

We can rewrite this function as $y\left(x\right)=x^3(x+1{)}^{-2}$ and apply product rule,
$\frac{dy}{dx}=(x+1{)}^{-2}\frac{d}{dx}(x^3)+x^3\frac{d}{dx}(x+1{)}^{-2}\frac{dy}{dx}=(x+1{)}^{-2}.3x^2+x^3(-2\left)\right(x+1{)}^{-3}\frac{dy}{dx}=\frac{3x^2}{(x+1{)}^2}-\frac{2x^3}{(x+1{)}^3}$