Question

How do you differentiate `x * (4-x^2)^(1/2)`?

Answer 1

I found: `2(2-x^2)/(sqrt(4-x^2))`

Explanation:

You can use the Product Rule to deal with the product between the two functions and the Chain Rule to deal with the `()^(1/2)`:
`y'=1*(4-x^2)^(1/2)+1/2x(4-x^2)^(1/2-1)*(-2x)=`
`=(4-x^2)^(1/2)-x^2(4-x^2)^(-1/2)=`
Considering that: `(4-x^2)^(1/2)=sqrt(4-x^2)` and `(4-x^2)^(-1/2)=1/sqrt(4-x^2)`
You get:
`sqrt(4-x^2)-x^2/(sqrt(4-x^2))=(4-x^2-x^2)/(sqrt(4-x^2))=`
`=(4-2x^2)/(sqrt(4-x^2))=2(2-x^2)/(sqrt(4-x^2))`