Question

How do you differentiate `f(x)=x^2 / (1-x)^(1/2)` using the quotient rule?

Answer 1

The quotient rule states that for a function `y=f(x)/g(x)`, its derivative is `(dy)/(dx)=(f'(x)g(x)-f(x)g'(x))/g(x)^2`

Explanation:

Also, here we'll need the chain rule when derivating the second term. The chain rule states that `dy/dx=dy/(du)(du)/dx`. Let's rename `u=1-x`. Done, let's start.

`(dy)/(dx)=(2x(1-x)^(1/2)-x^2(1/(2u^(1/2)))*(-1))/((1-x)^(1/2))^2`

`(dy)/(dx)=(2x(1-x)^(1/2)+x^2/(2(1-x)^(1/2)))/(1-x)`

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

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