\begin{frame} \frametitle{Derivatives} \begin{block}{} The \emph{derivative of a function $f$ at a number $a$}, denoted $f'(a)$, is \begin{talign} f'(a) = \lim_{h\to 0} \frac{f(a+h) - f(a)}{h} \end{talign} if the limit exits. \end{block} \pause An equivalent way of defining the derivative (take $x = a+h$): \begin{talign} f'(a) = \lim_{x\to a} \frac{f(x) - f(a)}{x - a} \end{talign}\vspace{-1ex} \pause\medskip \begin{block}{} The tangent line to $f$ at point $(a,f(a))$ is the line through $(a,f(a))$ with slope $f'(a)$, the derivative of $f$ at $a$. \end{block} \pause \begin{exampleblock}{} Find an equation of the tangent to $f(x) = x^2 - 8x + 9$ at $(3,-6)$. \pause\smallskip We know $f'(a) = 2a-8$\pause, and thus $f'(3) = -2$.\\\pause Hence \quad$y+6 = -2(x-3)$\quad\pause,that is, \quad$y = -2x$ \end{exampleblock} \end{frame}