derivative of $e^x$ using the limit definition

Question

Looking to prove $ f(x) = e^x =f'(x) $ using the limit definition of the derivative. So far I have gotten to here:

$e^x * \lim\limits_{h \to 0} \frac{e^h-1}{h}$

Any hints on what I can do next to get that entire limit to $1$?

I can't use l'hopital's rule, otherwise that'd defeat the purpose.

Answer 1

$$\large e=\lim_{\Delta x \to 0 }(1+\Delta x)^{\frac{1}{\Delta x}}\space (definition)\\ e^x=\lim_{\Delta x \to 0 }(1+\Delta x)^{\frac{x}{\Delta x}}\\ f'(x)=\lim_{\Delta x \to 0 }\frac{f(x+\Delta x)-f(x)}{\Delta x }=\\ \large \lim_{\Delta x \to 0 }\frac{(1+\Delta x)^{\frac{x+\Delta x}{\Delta x}}-(1+\Delta x)^{\frac{x}{\Delta x}}}{\Delta x }=\\ \large \lim_{\Delta x \to 0 }\frac{(1+\Delta x)^{\frac{x}{\Delta x}}((1+\Delta x)^{\frac{\Delta x}{\Delta x}}-1)}{\Delta x }=\\ \large \lim_{\Delta x \to 0 }\frac{(1+\Delta x)^{\frac{x}{\Delta x}}((1+\Delta x)^1-1)}{\Delta x }=\\ \large \lim_{\Delta x \to 0 }\frac{(1+\Delta x)^{\frac{x}{\Delta x}}(\Delta x)}{\Delta x }=\\ \large \lim_{\Delta x \to 0 }(1+\Delta x)^{\frac{x}{\Delta x}}=\\e^x$$

Answer 2

Derivative of a function is not that difficult to calculate provided you know the definition of the function very well. Most students who start to learn calculus are aware only of the definition of polynomials, rational functions and to some extent algebraic functions. Thanks to many crappy textbooks they are led to believe that they also know the definitions of circular, logarithmic and exponential functions. Proper definitions of these functions is not possible without a proper theory of real numbers and some amount of real-analysis or calculus.

One way out (especially suitable for beginners in calculus) is not to define these complicated functions, but rather state their most common properties (including certain limits associated with them) without proof and then using these properties the derivative of these functions can be calculated. For the case of $f(x) =e^{x}$ we need to know two properties $$e^{a+b} =e^{a} e^{b}, \, \lim_{x\to 0}\frac{e^{x}-1}{x}=1$$ and using these you can easily show that the derivative of $e^{x} $ is $e^{x} $ itself.

Later when you have attained some maturity in calculus you can very well learn a proper definition of $e^{x} $ using which you can prove the properties mentioned above. Again there are many possible options to define these functions and anyone can be chosen. My own preferred definition is given by the equation $$e^{x} =\lim_{n\to\infty}\left(1+\frac{x}{n}\right)^{n}$$ and you can study the development of these functions based on this definition in this post.