Question

How do you find the derivative of `y=e^x`?

Answer 1

This is one of the favorite function to take the derivatives of.
`y'=e^x`

If you wish to find this derivative by the limit definition, then here is how we find it. First, we have to know the following property of `e`:
`lim_{h to 0}{e^h-1}/{h}=1`.
(Note: This means that the slope of `y=e^x` at `x=0` is `1`.)

By the limit definition of the derivative, we have
#y'=lim_{h to 0}{e^{x+h}-e^x}/h =lim_{h to 0}{e^x cdot e^h-e^x}/h#
by factoring out `e^x`,
`=lim_{h to 0}{e^x(e^h-1)}/h=e^x lim_{h to 0}{e^h-1}/h`
by the property of `e` mentioned above,
`=e^x cdot 1=e^x`

Hence, the derivative of `e^x` is itself.