Opgave
De afgeleide bepalen van:
$ \eqalign{f(x) = \frac{{(3 - x)\sqrt x }} {3}} $
Aanpak
Schrijf $f(x)$ ala: $ f(x) = \frac{1} {3}(3 - x) \cdot \sqrt x $
Met de productregel:
$
\eqalign{
& g(x) = \frac{1}
{3}(3 - x) \cr
& h(x) = \sqrt x \cr
& f'(x) = g'(x) \cdot h(x) + g(x) \cdot h'(x) \cr}
$
Uitwerking
$ \eqalign{ & f(x) = \frac{1} {3}(3 - x) \cdot \sqrt x \cr & f'(x) = \frac{1} {3} \cdot - 1 \cdot \sqrt x + \frac{1} {3}(3 - x) \cdot \frac{1} {{2\sqrt x }} \cr & f'(x) = - \frac{1} {3}\sqrt x + \frac{{3 - x}} {{6\sqrt x }} \cr & f'(x) = - \frac{1} {3}\sqrt x \cdot \frac{{2\sqrt x }} {{2\sqrt x }} + \frac{{3 - x}} {{6\sqrt x }} \cr & f'(x) = \frac{{ - 2x}} {{6\sqrt x }} + \frac{{3 - x}} {{6\sqrt x }} \cr & f'(x) = \frac{{ - 2x + 3 - x}} {{6\sqrt x }} \cr & f'(x) = \frac{{ - 3x + 3}} {{6\sqrt x }} = \cr & f'(x) = \frac{{ - x + 1}} {{2\sqrt x }} \cr & {\text{of}}\,\,\,{\text{ook}}: \cr & f'(x) = \frac{{1 - x}} {{2\sqrt x }} \cr} $
