\begin{frame} \frametitle{Higher Derivatives} If $f$ is a function, the derivative $f'$ is also a function. \pause\medskip Thus we can compute the derivative of the derivative: \begin{talign} (f')' = f'' \end{talign} \pause The function $f''$ is called \emph{second derivative} of $f$. \pause\smallskip % In Leibnitz notation the second derivative of $y = f(x)$ is written as: % \begin{talign} % \frac{d}{dx} \left(\frac{dy}{dx} \right) \;=\; \frac{d^2y}{dx^2} % \end{talign} \begin{exampleblock}{} Let $f(x) = x^3 - x$. Find $f''(x)$. \pause\medskip We have seen $f'(x) = 3x^2 - 1$. Thus \begin{talign} f''(x) &= (f')'(x) \mpause[1]{= \lim_{h\to 0} \frac{f'(x+h) - f'(x)}{h}}\\ &\mpause[2]{= \lim_{h\to 0} \frac{[3(x+h)^2 - 1] - [3x^2 - 1]}{h}}\\ &\mpause[3]{= \lim_{h\to 0} \frac{3x^2 + 6xh + 3h^2 - 1 - 3x^2 + 1}{h}}\\ &\mpause[4]{= \lim_{h\to 0} \frac{6xh + 3h^2}{h}} \mpause[5]{= \lim_{h\to 0} (6x + 3h)} \mpause[6]{= 6x} \end{talign} \end{exampleblock} \end{frame}