TeX source:

\mathrm{f}^{\prime}(\mathrm{x})=\lim\limits_{h \rightarrow 0} \frac{f(x+h)-f(x)}{h}=\lim\limits_{h \rightarrow 0} \frac{\left\{(x+h)^{2}+8\right\}-\left\{x^{2}+8\right\}}{h}=\lim\limits_{h \rightarrow 0} \frac{2 x h+h^{2}}{h}=\lim\limits_{h \rightarrow 0} 2 x+h=2 x \cdot \mathrm{f}^{\prime}(3)=6