# Exploration: Exponential Functions and Derivatives - The Derivative and Tangent Lines

Author(s):
Tom Leathrum

With the formula for the derivative of f(x)=ex giving f '(x)=ex, the derivative can be used to find slopes of tangent lines to the graph of the function f(x)=ex. At a point (x0,y0) on the graph of f(x) (so that y0=f(x0)), the line tangent to the graph will have slope m=f '(x0). Plugging into the Point-Slope form equation for a line, then, the equation for the tangent line at (x0,y0) will be:

y-y0=m(x-x0)
y-f(x0)=f '(x0) (x-x0)

This is the equation used to find the tangent lines to the graph in the applet below. Using the functions f(x)=ex and f '(x)=ex, the applet shows the tangent line for any given value of x0 in the graph, starting below with x0=1.

Other values for x0 (shown in the "x=" field in the applet) can be chosen either by entering a new value into the "x=" field or by clicking and dragging the mouse on the graph.

Tom Leathrum, "Exploration: Exponential Functions and Derivatives - The Derivative and Tangent Lines," Convergence (October 2004)

## JOMA

Journal of Online Mathematics and its Applications