Opgave 1
$
\eqalign{
& f(x) = x^3 \cdot e^{x^2 } \cr
& f'(x) = \left[ {x^3 } \right]^' \cdot e^{x^2 } + x^3 \cdot \left[ {e^{x^2 } } \right]^' \,\,\,\,\left( {productregel} \right) \cr
& f'(x) = 3x^2 \cdot e^{x^2 } + x^3 \cdot e^{x^2 } \cdot 2x\,\,\,\,\,\,\left( {kettingregel} \right) \cr
& f'(x) = 3x^2 \cdot e^{x^2 } + 2x^4 \cdot e^{x^2 } \cr
& f'(x) = \left( {3 + 2x^2 } \right)x^2 e^{x^2 } \,\,\,\,\,\,\,\left( {factor\,\,buiten\,\,haakjes\,\,halen} \right) \cr}
$
Opgave 2
$
\eqalign{
& f(x) = e^{\sin 2x} \cr
& f'(x) = e^{\sin 2x} \cdot \left[ {\sin 2x} \right]^' \,\,\,\left( {kettingregel} \right) \cr
& f'(x) = e^{\sin 2x} \cdot \cos 2x \cdot 2\,\,\,\,\left( {kettingregel} \right)\, \cr
& f'(x) = 2\cos 2x \cdot e^{\sin 2x} \cr}
$
Opgave 3
$
\eqalign{
& f(x) = \frac{{2e^x - 3}}
{{e^x + 1}} \cr
& f'(x) = \frac{{\left[ {2e^x - 3} \right]^' \cdot \left( {e^x + 1} \right) - \left( {2e^x - 3} \right) \cdot \left[ {e^x + 1} \right]}}
{{\left( {e^x + 1} \right)^2 }}^' \,\,\,\,\,\,\left( {quotiëntregel} \right) \cr
& f'(x) = \frac{{2e^x \cdot \left( {e^x + 1} \right) - \left( {2e^x - 3} \right) \cdot e^x }}
{{\left( {e^x + 1} \right)^2 }}\, \cr
& f'(x) = \frac{{2e^{2x} + 2e^x - \left( {2e^{2x} - 3e^x } \right)}}
{{\left( {e^x + 1} \right)^2 }}\,\,\,\,\,\,\left( {haakjes\,\,wegwerken} \right) \cr
& f'(x) = \frac{{2e^{2x} + 2e^x - 2e^{2x} + 3e^x }}
{{\left( {e^x + 1} \right)^2 }}\,\,\,\,\,\,\left( {gelijksoortige\,\,termen\,\,samennemen} \right) \cr
& f'(x) = \frac{{5e^x }}
{{\left( {e^x + 1} \right)^2 }} \cr}
$
