Directly from the definition of exponential function You first demonstrate the 'fundamental limit'...
... and using (1) it is easy to demonstrate that is...
Is it actually possible to derive the derivative of from first principles?
That is, to prove .
How do you go from there?
Or is it impossible because we actually define as a function whose derivative is itself?
May be that most of You don't agree with me but I'm strongly convinced that the only 'rigorous' mathematical arguments are based on the four elementary operations... so that, in my opinion of course, the only 'rigorous' definition of 'exponential function' is the definition given by Leonhard Euler two and half centuries ago...
I don't see that the limit definition is significantly closer to a definition based on "the four elementary operations... " than the differential equation definition or the inverse of the natural logarithm, etc...
Clearly, how you find the derivative of any function depends on how you define that function.
It is perfectly valid to define . From that the fundamental theorem of calculus gives
We then define to be the inverse function to ln(x). If then so and then
i think it is perfectly "rigorous" to define in terms of power series (after all, we are dealing with real numbers, so convergence comes with the territory).
but this is the long way around, you have to define an infinite series, and what convergence for a series means, and then to show the usual power series for converges for all real x. then, you need to show that term-by-term differentiation is justified. having done all that, the proof of the derivative is easy.
the next best thing, is to define ln(x) as , and then define . but then, to exhibit the derivative, one has to invoke the inverse function theorem (or dx/dy = 1/(dy/dx), which is what one does in practice, but is in my opinon, a little imprecise).
as for the proof of the so-called multiplicative property of ln(x) chisigma requested, here it is:
fix y > 0, and define for x > 0:
f(x) = ln(xy). then f'(x) = (ln'(xy))(xy)' (the chain rule)
= (1/(xy))(y) = 1/x. hence f' = ln', so f and ln differ by a constant, say c:
f(x) = ln(x) + c, for all x > 0. in particular, this holds for x = 1:
so ln(y) = ln(1y) = f(1) = ln(1) + c = c.
since y can be any positive real number,
ln(xy) = ln(x) + ln(y) for all x,y > 0.
Thanks for replying.
Looking at posts #3 and #4 by Prove it and chisigma respectively, which one came first? The definition of or the defintion of as a function whose derivative is itself? Since if you use either one definition, you can always prove the other.
I think this is what caused my confusion.
Very well, ChiSigma.
If we define , then it is immediate that ln(x) is defined, continuous, and differentiable for all x> 0. It is also immediate that .
For x> 0, 1/x is also greater than 0 and we have . Make the substitution u= xt so that t= u/x and dt= du/x. Also, when t= 1, u= x, when t= 1/x, u= 1. Then .
For x and y> 0, xy> 0 and we have . Make the substitution u= x/t so that t= xu and dt= xdu. Also, when x= 1, u= 1/x, when t= xy, u= y. Then .
For x> 0 and y any real number, is also positive and we have . If y is not 0, we can make the substitution so that and . Also when t= 1, u= 1 and when , . .
If y= 0, then so that so is true for any y.
Since is differentiable for all positive x, we can apply the mean value theorem on, say, the interval from x= 1 to x= 2. where . Then . That is, .
That is important for the following reason: If X is any positive real number, then . That is, ln(x) is not bounded above. Since its derivative, 1/x, is always positive, ln(x) is an increasing function and since it is not bounded above, goes to infinity as x goes to infinity. From ln(1/x)= -ln(x), it is easy to see that ln(x) goes to negative infinity as x goes to 0. That shows that ln(x) maps the set of all positive real number "one to one and onto" the set of all real numbers and so has an inverse. We define exp(x) to be that inverse function.
One thing remains to be shown. If y= exp(x), then x= ln(y). If x is not 0, we can divide by it: . Going back to the exponential, so that . Of course, if x= 0, so . That is, the "exp" function, defined as the inverse function to ln(x) really is some number to the x power. If we now define e to be exp(1) (the number whose natural logarithm is 1), then we have .