# What is the derivative of x*sqrt(4-x)?

Nov 3, 2016

The answer is $= \frac{8 - 3 x}{2 \sqrt{4 - x}}$
Use $\left(u v\right) ' = u ' v + u v '$
$u = x$$\implies$$u ' = 1$
$v = \sqrt{4 - x}$$\implies$$v ' = \frac{1}{2 \sqrt{4 - x}} \cdot - 1$
$\left(x \cdot \sqrt{4 - x}\right) ' = 1 \cdot \sqrt{4 - x} - \frac{x}{2 \sqrt{4 - x}}$
$= \frac{2 \left(4 - x\right) - x}{2 \sqrt{4 - x}} = \frac{8 - 2 x - x}{2 \sqrt{4 - x}} = \frac{8 - 3 x}{2 \sqrt{4 - x}}$