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

1 Answer
Mar 5, 2018

Answer:

#(2(2x+1)*(2))*((-14x)/(x^2-2)^2)#

Explanation:

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

#(2x+1)^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'(g(x)) * g'(x)#

This will give us

#2(2x+1)*(2)#

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

#(((2x)*(x^2-2))-((x^2+5)*(2x))) / (x^2-2)^2 #

After you simplify it should look like

#((2x^3-4x)-(2x^3+10x))/(x^2-2)^2#

Then you end up with

#(-14x)/(x^2-2)^2#

Finally you can combine the two to get your final answer

#(2(2x+1)*(2)) * ((-14x)/(x^2-2)^2)#