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

Dec 14, 2017

$- \frac{2 \cdot \left(- 2 + {x}^{2}\right)}{\sqrt{4 - {x}^{2}}}$

#### Explanation:

Using the product rule:

$f ' \left(a \cdot b\right) = b \cdot f ' \left(a\right) + a \cdot f ' \left(b\right)$

Let $\textcolor{w h i t e}{88} a = x$

Let $\textcolor{w h i t e}{88} b = {\left(4 - {x}^{2}\right)}^{\frac{1}{2}}$

$f ' \left(a\right) = 1$

$f ' \left(b\right) = \frac{1}{2} {\left(4 - {x}^{2}\right)}^{- \frac{1}{2}} \cdot - 2 x = - x {\left(4 - {x}^{2}\right)}^{- \frac{1}{2}}$

$f ' \left(a \cdot b\right) = {\left(4 - {x}^{2}\right)}^{\frac{1}{2}} \cdot 1 + x \cdot - x {\left(4 - {x}^{2}\right)}^{- \frac{1}{2}}$

$\to = {\left(4 - {x}^{2}\right)}^{\frac{1}{2}} + \frac{- {x}^{2}}{4 - {x}^{2}} ^ \left(\frac{1}{2}\right) = - \frac{2 \cdot \left(- 2 + {x}^{2}\right)}{\sqrt{4 - {x}^{2}}}$