How can I prove that the derivative of e^x is e^x?

Answer 1

the best way to appr this problem is to apply the first principle i.e.
f'(x)= Lt [f(x+h)-f(x)]/h
h->0
now putting f(x)=e^x we have
f'(x)= Lt [e^(x+h)-e^x]/h
h->0
=Lt [e^x(e^h-1)]/h
h->0

now (e^h-1)/h where h tends to 0 is 1
hence f'(x) = e^x
therefore the derivative of e^x is e^x

Answer 2

This depends strongly on how you define the number e and the function e^x. There are three main approaches:

1) Calculate the derivative of a^x for a general x from the limit definition of the derivative. It turns out to be C*a^x where
C=lim [a^h -1]/h
as h->0. This limit represents the slope of the tangent line to y=a^x at x=0. Now, this slope varies continuously as a changes. For a=2, it is about .693 and for a=3 it is about 1.098. So, for some number between 2 and 3, the limit will be 1. Call this number e and get that the derivative of e^x is e^x.

2) define
ln x=int 1/t dt
where the integral goes from 1 to x. Then define exp(x) to be the inverse function of ln x (which is clearly increasing, so 1-1). By the rules for differentiating inverse functions, the derivative of exp(x) will be exp(x). Then define e=exp(1) and show the rulesof exponents work for exp(x), showing e^x=exp(x).

3) Define
exp(x)=sum (1/n!) x^n
where n goes from 0 to infinity. This is a power series with infinite radius of convergence and so can be differentiated term-by term.
It is then found that the derivative of exp(x) is exp(x). Also, directly from the power series, we can show exp(x+y)=exp(x)*exp(y). So again, defining e=exp(1) gives exp(x)=e^x.

Hope this helps!

Answer 3

Other than using derivative tables, that is the best, I think. Can you use the definitions of 1st derivative(slope of the function) and 2nd derivative(concavity) to prove that it matches? That doesn't seem mathematically sound for a proof though.

Answer 4

Actually, the definition of e^x is the function that solves the following differential equation: dy/dx=y, y(0)=1. So there you go. You don't need to prove it. It IS the definition. However, in order to understand that this definition is well defined, you need to know that the solution to the differential equation is unique which requires understanding fixed points of infinite dimesional banach spaces.

Answer 5

The Derivative Table
#21... d/dx(e^x) = e^x du/dx
check with taking the integal
Integral Table
# 11.... sum
integal e^x dx = e^x

Answer 6

Use logarithmic differentiation:

y=e^x
lny=x*lne
d(lny)/dx=d(x*lne)/dx
y'/y=1
y'=y=e^x

:P

Answer 7

ummm...not sure about the question but it is the same as (e x e x e x e x e...) x times