\eqalign{
& Als\,\,f(x) = \frac{{g(x)}}
{{h(x)}}\,\,dan: \cr
& f'(x) = \frac{{g'(x) \cdot h(x) - g(x) \cdot h'(x)}}
{{\left( {h(x)} \right)^2 }} \cr}
Voorbeeld 1
\eqalign{
& Als\,\,f(x) = \frac{{2x}}
{{x^2 - 2}}\,\,dan: \cr
& f'(x) = \frac{{2 \cdot \left( {x^2 - 2} \right) - 2x \cdot 2x}}
{{\left( {x^2 - 2} \right)^2 }} \cr
& f'(x) = \frac{{2x^2 - 4 - 4x^2 }}
{{\left( {x^2 - 2} \right)^2 }} \cr
& f'(x) = \frac{{ - 2x^2 - 4}}
{{\left( {x^2 - 2} \right)^2 }} \cr}
Voorbeeld 2
\eqalign{
& f(t) = \frac{{1 - 3t - 3t^2 }}
{{3t^2 - 2t^3 }} \cr
& f'(t) = \frac{{\left( { - 3 - 6t} \right)\left( {3t^2 - 2t^3 } \right) - \left( {1 - 3t - 3t^2 } \right)\left( {6t - 6t^2 } \right)}}
{{\left( {3t^2 - 2t^3 } \right)^2 }} \cr
& f'(t) = ... \cr}
Voorbeeld 3
\eqalign{
& f(x) = \frac{x}
{{\left( {2x - 1} \right)^2 }} \cr
& f'(x) = \frac{{1 \cdot \left( {2x - 1} \right)^2 - x \cdot 2\left( {2x - 1} \right) \cdot 2}}
{{\left( {\left( {2x - 1} \right)^2 } \right)^2 }} \cr
& f'(x) = \frac{{\left( {2x - 1} \right)^2 - 4x\left( {2x - 1} \right)}}
{{\left( {\left( {2x - 1} \right)^2 } \right)^2 }} \cr
& f'(x) = \frac{{2x - 1 - 4x}}
{{\left( {2x - 1} \right)^3 }} \cr
& f'(x) = \frac{{ - 2x - 1}}
{{\left( {2x - 1} \right)^3 }} \cr
& f'(x) = \frac{{2x + 1}}
{{\left( {1 - 2x} \right)^3 }} \cr}
Voorbeeld 4
\eqalign{
& f(x) = 12 \cdot \frac{{\left( {x^2 - 1} \right)}}
{{x^2 + 12}} \cr
& f'(x) = 12 \cdot \frac{{\left[ {\left( {x^2 - 1} \right)} \right]^| \cdot \left( {x^2 + 12} \right) - \left( {x^2 - 1} \right) \cdot \left[ {x^2 + 12} \right]^| }}
{{\left( {x^2 + 12} \right)^2 }} \cr
& f'(x) = 12 \cdot \frac{{2x \cdot \left( {x^2 + 12} \right) - \left( {x^2 - 1} \right) \cdot 2x}}
{{\left( {x^2 + 12} \right)^2 }} \cr
& f'(x) = 12 \cdot \frac{{2x\left\{ {\left( {x^2 + 12} \right) - \left( {x^2 - 1} \right)} \right\}}}
{{\left( {x^2 + 12} \right)^2 }} \cr
& f'(x) = 12 \cdot \frac{{2x\left\{ {x^2 + 12 - x^2 + 1} \right\}}}
{{\left( {x^2 + 12} \right)^2 }} \cr
& f'(x) = 12 \cdot \frac{{2x \cdot 13}}
{{\left( {x^2 + 12} \right)^2 }} \cr
& f'(x) = \frac{{312x}}
{{\left( {x^2 + 12} \right)^2 }} \cr}
Voorbeeld 5
\eqalign{
& f(x) = \frac{{a^2 - x^2 }}
{{a^2 + x^2 }} \cr
& f'(x) = \frac{{ - 2x\left( {a^2 + x^2 } \right) - \left\{ {\left( {a^2 - x^2 } \right) \cdot 2x} \right\}}}
{{\left( {a^2 + x^2 } \right)^2 }} \cr
& f'(x) = \frac{{ - 2a^2 x - 2x^3 - \left\{ {2a^2 x - 2x^3 } \right\}}}
{{\left( {a^2 + x^2 } \right)^2 }} \cr
& f'(x) = \frac{{ - 2a^2 x - 2x^3 - 2a^2 x + 2x^3 }}
{{\left( {a^2 + x^2 } \right)^2 }} \cr
& f'(x) = \frac{{ - 4a^2 x}}
{{\left( {a^2 + x^2 } \right)^2 }} \cr}
F.A.Q.
