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

1 Answer
Nov 16, 2017

Answer:

#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#