# Math Insight

### Applet: The derivative of an exponential function

Illustration of how the derivative of the exponential function is a multiple of the function, where that multiple is the derivative at zero. The graph of the function $f(x)=b^x$, where you can enter a value for $b$, is shown by the thick blue curve. The graph of its derivative $f'(x)$ is shown by the thin green curve, and the ratio $f'(x)/f(x)$ is shown by the horizontal gray line. The value of the function and its derivative evaluated at $x_0$ are displayed at the left and illustrated by the blue and green points on the curves. You can change $x_0$ by typing in a new value or dragging one of the points. The red line is the tangent line at $x_0$ with slope $f'(x_0)$. The ratio $f'/f$ evaluate at $x_0$ is also displayed at the left. If you check the “more options” check box, the function changes to $f(x)=c b^{kx}$ and two new boxes appear in which you can change the values of $c$ and $k$.

You can use the buttons at the top to zoom in and out as well as pan the view.

Applet file: derivative_exponential_function.ggb