What is the derivative of y=(2x+1)^2((x^2+5)/(x^2-2))?

Mar 5, 2018

$\left(2 \left(2 x + 1\right) \cdot \left(2\right)\right) \cdot \left(\frac{- 14 x}{{x}^{2} - 2} ^ 2\right)$

Explanation:

To begin, you want to use the chain rule for the first portion

${\left(2 x + 1\right)}^{2}$

The chain rule is done by finding the derivative of the outside, keeping the inside, and after that you will multiply it by the derivative of the inside.

$f ' \left(g \left(x\right)\right) \cdot g ' \left(x\right)$

This will give us

$2 \left(2 x + 1\right) \cdot \left(2\right)$

Next you want to find the derivative of the next function using the quotient rule.

((f'(x)g(x))-(f(x)g'(x)))/(g(x))^2

It will look like

$\frac{\left(\left(2 x\right) \cdot \left({x}^{2} - 2\right)\right) - \left(\left({x}^{2} + 5\right) \cdot \left(2 x\right)\right)}{{x}^{2} - 2} ^ 2$

After you simplify it should look like

$\frac{\left(2 {x}^{3} - 4 x\right) - \left(2 {x}^{3} + 10 x\right)}{{x}^{2} - 2} ^ 2$

Then you end up with

$\frac{- 14 x}{{x}^{2} - 2} ^ 2$

$\left(2 \left(2 x + 1\right) \cdot \left(2\right)\right) \cdot \left(\frac{- 14 x}{{x}^{2} - 2} ^ 2\right)$