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Uitwerkingen

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}


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