With the formula for the derivative of
*f*(*x*)=*e*^{x} giving
*f* '(*x*)=*e*^{x},
the derivative can be used to find slopes of tangent lines to the graph
of the function *f*(*x*)=*e*^{x}.
At a point (*x*_{0},*y*_{0}) on the
graph of *f*(*x*)
(so that *y*_{0}=*f*(*x*_{0})),
the line tangent to the graph will have slope
*m*=*f* '(*x*_{0}). Plugging into the
Point-Slope form equation for a line, then, the equation for the tangent
line at (*x*_{0},*y*_{0}) will be:

*y*-*y*_{0}=*m*(*x*-*x*_{0})

*y*-*f*(*x*_{0})=*f* '(*x*_{0})
(*x*-*x*_{0})
This is the equation used to find the tangent lines to the graph in the
applet below. Using the functions
*f*(*x*)=*e*^{x} and
*f* '(*x*)=*e*^{x}, the applet shows the
tangent line for any given value of *x*_{0} in the graph, starting
below with *x*_{0}=1.

How to use this applet

Other values for *x*_{0} (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.

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