Question

What is the derivative of `x * ((4-x^2)^(1/2))`?

Answer 1

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

Explanation:

`y=x*sqrt(4-x^2)`

I'll differentiate using the product and chain rules.

Product rule
`y=f(x)*g(x)`

`y'=f'(x)g(x) +g'(x)f(x)`

Chain rule

`y=f(g(x))`

`y' = f'(g(x))*g'(x)`

Differentiate each part
`f(x)=x`

`f'(x)=1`

`g(x)=sqrt(4-x^2)`

( `y=f(g(x)) => y' = f'(g(x))*g'(x)`)

`g'(x)=1/2(4-x^-2)^(-1/2)(-2x)`

so using the product rule

`y'=f'(x)g(x) +g'(x)f(x)`

`y'=sqrt(4-x^2) +(1/2(4-x^-2)^(-1/2)(-2x))(x)`

`y'=sqrt(4-x^2) +(-x^2)/sqrt(4-x^2)`

`y'=(4-x^2 -x^2)/sqrt(4-x^2)`

`y'=(-2x^2+4)/(sqrt(-x^2+4))`

which is the derivative of the function.