The derivative of ex
The function f(x)=ex is quite peculiar: it is the only function whose derivative is
itself.
ddx(ex)=ex. The
derivative of ex is ex.
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Perhaps (ex)′ is now your favorite derivative.
DO: Find the derivative
of g(x)=5⋅ex.
What follows is the reasoning behind why (ex)′=ex.
Please enjoy the following work and video if you are interested in
this unusual function.
We have f′(x)=limh→0f(x+h)−f(x)h=limh→0ex+h−exh=limh→0exeh−exh=ex⋅(limh→0eh−1h). The last step was
made possible by the fact that ex doesn't depend on h.
We are left with limh→0eh−1h. This
limit is 1, which you can guess by exploring this limit
graphically or numerically. Thus, the derivative
of ex is ex.
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