# What is the second derivative of f(x)=x^2/(x+3) ?

##### 1 Answer
Nov 19, 2016

Differentiate once, and then differentiate again.

By the quotient rule:

$f ' \left(x\right) = \frac{2 x \left(x + 3\right) - {x}^{2} \left(1\right)}{x + 3} ^ 2$

$f ' \left(x\right) = \left(2 {x}^{2} + 6 x - {x}^{2}\right) / {\left(x + 3\right)}^{2}$

$f ' \left(x\right) = \frac{{x}^{2} + 6 x}{{x}^{2} + 6 x + 9}$

Differentiate once more.

$f ' ' \left(x\right) = \frac{\left(2 x + 6\right) \left({x}^{2} + 6 x + 9\right) - \left(\left(2 x + 6\right) \left({x}^{2} + 6 x\right)\right)}{{x}^{2} + 6 x + 9} ^ 2$

You can simplify this further, but I'll leave the algebra up to you.

Hopefully this helps!