Question

How do you find f'(x) using the definition of a derivative for `f(x)=e^x`?

Answer 1

Refer to explanation

Explanation:

The definition of the first derivative is

`f'(x)=lim_(h->0) (f(x+h)-f(x))/(h)`

hence for `f(x)=e^x` we have that

#f'(x)=lim_(h->0) (e^(x+h)-e^x)/h=lim_(h->0) e^x*((e^h-1)/h)= e^x*lim_(h->0)((e^h-1)/h)=e^x#

Remarks

`lim_(h->0) ((e^h-1)/h)=1`