How do you use chain rule with a product rule to differentiate y = x*sqrt(1-x^2)?

1 Answer
Apr 1, 2015

If we selected a number for x and did the arithmetic, then the last operation would be multiplication. So, ultimately this is a product.

The derivative w.r.t. x of the first function is 1.

To find the derivative of the second function, we'll need the power rule (the derivative of the square root) and the chain rule. (Because we're not just taking square root of x.)

Using the product rule in the form: (FS)' = F'S+FS' we get:

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

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

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