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

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: $\left(F S\right) ' = F ' S + F S '$ we get:

$y ' = \left(1\right) \left(\sqrt{1 - {x}^{2}}\right) + \left(x\right) \left(\frac{1}{2 \sqrt{1 - {x}^{2}}} \cdot \left(- 2 x\right)\right)$

$= \sqrt{1 - {x}^{2}} - \frac{{x}^{2}}{\sqrt{1 - {x}^{2}}} = \frac{\sqrt{1 - {x}^{2}}}{1} - \frac{{x}^{2}}{\sqrt{1 - {x}^{2}}}$

$= \frac{\left(1 - {x}^{2}\right) - {x}^{2}}{\sqrt{1 - {x}^{2}}} = \frac{1 - 2 {x}^{2}}{\sqrt{1 - {x}^{2}}}$