{ \large f(x)\,=\,\frac{1}{2}\,{{x}^{2}}\,+\frac{1}{2}}
{\large \begin{array}{l}m=\,\underset{h\to 0}{\mathop{\lim }}\,\frac{f(1+h)-f(1)}{h}\,=\,\underset{h\to 0}{\mathop{\lim }}\,\,\frac{\overbrace{\frac{1}{2}\cdot {{\left( 1+h \right)}^{2}}\,+\frac{1}{2}\,}^{f({{x}_{0}}+h)}-\,\overbrace{\left( \frac{1}{2}\,\cdot {{1}^{2}}\,+\frac{1}{2} \right)}^{f({{x}_{0}})}}{h}\,=\,\underset{h\to 0}{\mathop{\lim }}\,\,\frac{\overbrace{0,5\cdot \left( 1+2h+{{h}^{2}} \right)+0,5}^{f({{x}_{0}}+h)}-\overbrace{1}^{f({{x}_{0}})}}{h}\\\\Klammer\,\,aufl\ddot{o}sen\underset{h\to 0}{\mathop{\lim }}\,\,\frac{0,5+h+\frac{{{h}^{2}}}{2}-0,5}{h}\,=\,\underset{h\to 0}{\mathop{\lim }}\,\,\frac{h+\frac{{{h}^{2}}}{2}}{h}\\\\h\,\,ausklammern\underset{h\to 0}{\mathop{\lim }}\,\,\frac{h\,\cdot \left( 1+\frac{h}{2} \right)}{h}\,\,=\,\underset{h\to 0}{\mathop{\lim }}\,\,\left( 1+\frac{h}{2} \right)\\\\Wenn\,\,h\,\,gegen\,\,0\,\,strebt,\,\,dann\,\,strebt\,\,\frac{h}{2}\,\to \,0\\\\\underset{h\to 0}{\mathop{\lim }}\,\,\left( 1+\frac{h}{2} \right)\,=\,1\end{array} }