A2.5

determine, through investigation using technology, the graph of the derivative $f^{\prime }(x)$ or $\frac{dy}{dx}$ of a given exponential function [i.e., $f(x)=a^x$ $(a>0,a\ne 1)$] [e.g., by generating a table of values showing the instantaneous rate of change of the function for various values of $x$ and graphing the ordered pairs; by using dynamic geometry software to verify that when $f(x)=a^x$, $f^{\prime }(x)=kf(x)$], and make connections between the graphs of $f(x)$ and $f^{\prime }(x)$ or $y$ and $\frac{dy}{dx}$ [e.g., $f(x)$ and $f^{\prime }(x)$ are both exponential; the ratio $\frac{f^{\prime }(x)}{f(x)}$ is constant, or $f^{\prime }(x)=kf(x)$; $f^{\prime }(x)$ is a vertical stretch from the x-axis of $f(x)$] Sample problem: Graph, with technology, $f(x)=a^x$ $(a>0,a\ne 1)$ and $f^{\prime }(x)$ on the same set of axes for various values of $a$ (e.g., 1.7, 2.0, 2.3, 3.0, 3.5). For each value of $a$, investigate the ratio $\frac{f^{\prime }(x)}{f(x)}$ for various values of $x$, and explain how you can use this ratio to determine the slopes of tangents to $f(x)$.