Question

What is the derivative of `f(x) = x(sqrt( 1 - x^2))`?

Answer 1

`(df)/dx = sqrt(1-x^2) - x^2/(sqrt(1-x^2))`.

Explanation:

We will require the use of two rules: the product rule and the chain rule. The product rule states that:

`(d(fg))/dx` = `(df)/dx * g(x) + f(x) * (dg)/dx`.

The chain rule states that:

`(dy)/dx = (dy)/(du) (du)/dx`, where `u` is a function of `x` and `y` is a function of `u`.

Therefore,

`(df)/dx = (x)' * (sqrt(1-x^2)) + x * (sqrt(1-x^2))'`

To find the derivative of `sqrt(1-x^2)`, use the chain rule, with

`u = 1-x^2: (sqrtu)' = 1/(2sqrtu) * u'`

`= -(2x)/(2(sqrt(1-x^2))` `= -x/(sqrt(1-x^2))`.

Substituting this result into the original equation:

`(df)/dx = sqrt(1-x^2) - x^2/(sqrt(1-x^2))`.