Question

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

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

Answer 1

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

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