Question

How do you differentiate `f(x) = (x - 1)^4 /(x^2+2x)^5` using the chain rule?

Answer 1

`f^-1(x)=(4(x-1)^3 - (x-1)^4(10x+10)(x^2+2x)^3)/(x^2+2x)^6`

Explanation:

`((x-1)^4)/ ((x^2+2x)^5)`

First, find which is u and which is v.

`u = (x-1)^4`
`v=(x^2+2x)^5`

To differentiate both, you have to use the chain rule. Make the 4 the coefficient of the bracket. Then differentiate what is in the bracket, which is 1, and multiply it by 4. Remember to subtract one from the power, becoming 3.

`(du)/dx = 4*1(x-1)^3 -> 4(x-1)^3`

Repeat the same thing for v.

`(dv)/dx = 5(2x+2)(x^2+2x)^4`

Now use the quotient rule to find `f^-1(x)`

quotient rule = `(v(du)/dx - u(dv)/dx)/v^2`

`(dy)/dx= (4(x^2+2x)(x-1)^3 - (x-1)^4(10x+10)(x^2+2x)^4)/(x^2+2x)^7`

`f^-1(x)=(4(x-1)^3 - (x-1)^4(10x+10)(x^2+2x)^3)/(x^2+2x)^6`