# How do you differentiate f(x)=(2+x )sqrt(2-3x using the product rule?

Oct 4, 2016

$f ' \left(x\right) = u ' v + v ' u$
Let $u = 2 + x$ and $v = \sqrt{2 - 3 x}$
then $u ' = 1$ and $v ' = - 3$ x $\frac{1}{2} {\left(2 - 3 x\right)}^{- \frac{1}{2}}$(chain rule)
Subbing into $f ' \left(x\right) = u ' v + v ' u$,
$f ' \left(x\right) = 1 \left(\sqrt{2 - 3 x}\right) + \left(- \frac{3}{2}\right) {\left(2 - 3 x\right)}^{- \frac{1}{2}} \left(2 + x\right)$
$f ' \left(x\right) = \sqrt{2 - 3 x} - \frac{3 \left(2 + x\right)}{2 \sqrt{2 - 3 x}}$