Question

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

Answer 1

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!