This applet is designed to study the relationship between the graph of exponential functions and their derivatives.

The function graphed in red is the graph of the exponential function \(f(x) = a^x\), where the base value \(a\) can be changed using the slider below the graph. Also shown (in blue) is the tangent line to the graph at \(x = 0\) and, below the slider, the value of the slope \(m\) of the tangent line. This slope is the derivative of the function at \(x = 0\), \(f'(0)\). By moving the slider, we can see how the base affects the graph of the function and its derivative at \(x = 0\).