\begin{frame} \frametitle{Exam Task from 2005} \begin{exampleblock}{} Using the definition of derivative, find $f'(x)$, where $f(x) = \sqrt{2x}$. \pause \begin{talign} f'(x) &= \lim_{h\to 0} \frac{f(x+h) - f(x)}{h}\\ &\mpause[1]{= \lim_{h\to 0} \frac{\sqrt{2x+2h} - \sqrt{2x}}{h}}\\ &\mpause[2]{= \lim_{h\to 0} \left( \frac{\sqrt{2x+2h} - \sqrt{2x}}{h} \cdot \frac{\sqrt{2x+2h} + \sqrt{2x}}{\sqrt{2x+2h} + \sqrt{2x}} \right)}\\ &\mpause[3]{= \lim_{h\to 0} \left( \frac{2x+2h - 2x}{h\cdot (\;\sqrt{2x+2h} + \sqrt{2x}\;)} \right)}\\ &\mpause[4]{= \lim_{h\to 0} \left( \frac{2}{\sqrt{2x+2h} + \sqrt{2x}} \right)}\\ &\mpause[5]{= \frac{2}{2\sqrt{2x}}} \mpause[6]{= \frac{1}{\sqrt{2x}}} \end{talign} \end{exampleblock} \end{frame}