How do you use the chain rule to differentiate #y=((x+2)/(x+1))^3#?

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Answer 1 · 2016-08-16T01:58:25

Let #u = v^3# and #v = (x + 2)/(x + 1)#.

We will have to differentiate both #y# and #u#. Our strategies for both tasks should be as follows:

•For u, differentiate using the power rule.
•For v, differentiate using the quotient rule.

#u' = 3v^2#

~~~~~~~~~~~~~~~~~~~~~~~~

#v' = (1 xx (x + 1) - 1 xx (x + 2))/(x + 1)^2#

#v' = (x + 1 - (x + 2))/(x + 1)^2#

#v' = -1/(x + 1)^2#

~~~~~~~~~~~~~~~~~~~~~~~~

As the chain rule states, the derivative of the entire function is the product of the derivative of #y# and the derivative of #u#.

Hence, #y' = 3v^2 xx -1/(x + 1)^2 = (-3((x + 2)/(x + 1))^2)/(x + 1)^2 = color(red)((-3(x + 2)^2)/(x + 1)^4)#

*The final answer is highlighted in #color(red)("red")#.

Hopefully this helps!