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

Aug 1, 2015

I found: $2 \frac{2 - {x}^{2}}{\sqrt{4 - {x}^{2}}}$

#### Explanation:

You can use the Product Rule to deal with the product between the two functions and the Chain Rule to deal with the ${\left(\right)}^{\frac{1}{2}}$:
$y ' = 1 \cdot {\left(4 - {x}^{2}\right)}^{\frac{1}{2}} + \frac{1}{2} x {\left(4 - {x}^{2}\right)}^{\frac{1}{2} - 1} \cdot \left(- 2 x\right) =$
$= {\left(4 - {x}^{2}\right)}^{\frac{1}{2}} - {x}^{2} {\left(4 - {x}^{2}\right)}^{- \frac{1}{2}} =$
Considering that: ${\left(4 - {x}^{2}\right)}^{\frac{1}{2}} = \sqrt{4 - {x}^{2}}$ and ${\left(4 - {x}^{2}\right)}^{- \frac{1}{2}} = \frac{1}{\sqrt{4 - {x}^{2}}}$
You get:
$\sqrt{4 - {x}^{2}} - {x}^{2} / \left(\sqrt{4 - {x}^{2}}\right) = \frac{4 - {x}^{2} - {x}^{2}}{\sqrt{4 - {x}^{2}}} =$
$= \frac{4 - 2 {x}^{2}}{\sqrt{4 - {x}^{2}}} = 2 \frac{2 - {x}^{2}}{\sqrt{4 - {x}^{2}}}$