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Math Help - Derivative of e^x from first principles

  1. #1
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    Derivative of e^x from first principles

    Hi guys,

    Is it actually possible to derive the derivative of e^x from first principles?

    That is, to prove \lim_{h \to 0}\frac{e^{x+h}-e^x}{h}=e^x.

    How do you go from there?

    Or is it impossible because we actually define e^x as a function whose derivative is e^x itself?

    Thanks.
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  2. #2
    MHF Contributor chisigma's Avatar
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    Re: Derivative of e^x from first principles

    Directly from the definition of exponential function You first demonstrate the 'fundamental limit'...

    \lim_{x \rightarrow 0} \frac{e^{x}-1}{x} = 1 (1)

    ... and using (1) it is easy to demonstrate that is...

    \lim_{h \rightarrow 0} \frac{e^{x + h}- e^{x}}{h} = \lim_{h \rightarrow 0} e^{x}\ \frac{e^{h}-1}{h} = e^{x} (2)

    Kind regards

    \chi \sigma
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  3. #3
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    Re: Derivative of e^x from first principles

    The function is DEFINED to be its own derivative. You CAN use this definition to evaluate the value of \displaystyle e though.

    \displaystyle \begin{align*} \lim_{h \to 0}\frac{e^{x + h} - e^x}{h} &= e^x \\ \lim_{h \to 0}\frac{e^xe^h - e^x}{h} &= e^x \\ \lim_{h \to 0}\frac{e^x(e^h - 1)}{h} &= e^x \\ e^x\lim_{h \to 0}\frac{e^h - 1}{h} &= e^x \\ \lim_{h \to 0}\frac{e^h - 1}{h} &= 1 \\ \lim_{h \to 0}(e^h - 1) &= \lim_{h \to 0}h \\ \lim_{h \to 0}e^h &= \lim_{h \to 0}(1 + h) \\ \lim_{h \to 0}(e^h)^{\frac{1}{h}} &= \lim_{h \to 0}(1 + h)^{\frac{1}{h}} \\ \lim_{h \to 0}(e) &= \lim_{h \to 0}(1 + h)^{\frac{1}{h}} \\ e &= \lim_{h \to 0}(1 + h)^{\frac{1}{h}} \end{align*}

    And if you let \displaystyle h = \frac{1}{n}, then \displaystyle n \to \infty as \displaystyle h \to 0, so we can write \displaystyle e as its more common limit form

    \displaystyle \begin{align*} e &= \lim_{h \to 0}(1 + h)^{\frac{1}{h}} \\ e &= \lim_{n \to \infty}\left(1 + \frac{1}{n}\right)^n  \end{align*}
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    MHF Contributor chisigma's Avatar
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    Re: Derivative of e^x from first principles

    To demonstrate that...

    \lim_{x \rightarrow 0} \frac{e^{x}-1}{x}= 1 (1)

    ... You start from the defintion of e...

    e= \lim_{\xi \rightarrow \infty} (1+\frac{1}{\xi})^{\xi} (2)

    ... and then set...

    e^{x} = 1+\frac{1}{\xi} \implies x= \ln (1+\frac{1}{\xi}) (3)

    ... so that is...

    \lim_{x \rightarrow 0} \frac{e^{x}-1}{x} = \lim_{\xi \rightarrow \infty} \frac{\frac{1}{\xi}}{\ln (1+\frac{1}{\xi})}= \lim_{\xi \rightarrow \infty} \frac{1}{\ln (1+\frac{1}{\xi})^{\xi}}= \frac{1}{\ln e} = 1 (4)

    Kind regards

    \chi \sigma
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    MHF Contributor chisigma's Avatar
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    Re: Derivative of e^x from first principles

    An important detail: to define e^{x} as the function that has itself as derivative is improper because any f(x)=c\ e^{x} , where c is an 'arbitrary constant', has itself as derivative...

    Kind regards

    \chi \sigma
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    Re: Derivative of e^x from first principles

    Quote Originally Posted by chisigma View Post
    An important detail: to define e^{x} as the function that has itself as derivative is improper because any f(x)=c\ e^{x} , where c is an 'arbitrary constant', has itself as derivative...

    Kind regards

    \chi \sigma
    The definition you mean is: e^x is the solution of the IVP

    f'(x)=f(x)

    with f(0)=1.

    Which gets around the problem of the constant.

    CB
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    MHF Contributor chisigma's Avatar
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    Re: Derivative of e^x from first principles

    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...

    e^{x}= \lim_{n \rightarrow \infty} (1+\frac{x}{n})^{n} (1)

    Kind regards

    \chi \sigma
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    Re: Derivative of e^x from first principles

    Quote Originally Posted by chisigma View Post
    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...

    e^{x}= \lim_{n \rightarrow \infty} (1+\frac{x}{n})^{n} (1)

    Kind regards

    \chi \sigma
    We may prefer one definition to another for aesthetic reasons but that does not make one better than the other in any absolute manner, nor does it make one more rigorous than the other.

    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...

    CB
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    Re: Derivative of e^x from first principles

    Clearly, how you find the derivative of any function depends on how you define that function.

    It is perfectly valid to define ln(x)= \int_1^x \frac{1}{t}dt. From that the fundamental theorem of calculus gives
    \frac{dln(x)}{dx}= \frac{1}{x}.

    We then define e^x to be the inverse function to ln(x). If y= e^x then x= ln(y) so \frac{dx}{dy}= \frac{1}{y} and then \frac{dy}{dx}= y= e^x
    Last edited by CaptainBlack; June 20th 2011 at 08:21 PM. Reason: replace [itex] tags by [tex]'s
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    MHF Contributor chisigma's Avatar
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    Re: Derivative of e^x from first principles

    Quote Originally Posted by HallsofIvy View Post
    Clearly, how you find the derivative of any function depends on how you define that function...

    It is perfectly valid to define \ln x= \int_1^x \frac{1}{t}dt...
    All right!... in that case I propose to all You the following 'easy little task': demonstrate on the basis of the defintion above that is...

    \ln (x_{1}\ x_{2}) = \ln x_{1} + \ln x_{2} (1)

    Kind regards

    \chi \sigma
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    Re: Derivative of e^x from first principles

    i think it is perfectly "rigorous" to define e^x 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 e^x 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 \ln(x) = \int_1^x \frac{1}{t} dt, and then define e^x = \ln^{-1}(x). 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.
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    Re: Derivative of e^x from first principles

    Thanks for replying.

    Looking at posts #3 and #4 by Prove it and chisigma respectively, which one came first? The definition of e=\lim_{n\to 0} (1+\frac{1}{n})^{n} or the defintion of e^x 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.
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    Re: Derivative of e^x from first principles

    Very well, ChiSigma.

    If we define ln(y)= \int_1^x \frac{dt}{t}, then it is immediate that ln(x) is defined, continuous, and differentiable for all x> 0. It is also immediate that \frac{d ln(x)}{dx}= \frac{1}{x}.

    For x> 0, 1/x is also greater than 0 and we have ln(1/x)= \int_1^{1/x}\frac{dt}{t}. 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 ln(1/x)= \int_x^1 \frac{x}{u}\frac{du}{x}= \int_x^1\frac{du}{u}= -\int_1^x \frac{du}{u}= -ln(x).

    For x and y> 0, xy> 0 and we have ln(xy)= \int_1^{xy}\frac{dt}{t}. 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 ln(xy}= \int_{1/x}^y \frac{1}{xu}xd= \int_{1/x}^y \frac{du}{u}= \int_{1/x}^1 \frac{du}{u}+ \int_1^y \frac{du}{u} = -\int_1^{1/x}\frac{du}{u}+ \int_1^y \frac{du}{u}= -ln(1/x)+ ln(y)= ln(x)+ ln(y).

    For x> 0 and y any real number, x^y is also positive and we have ln(x^y)= \int_1^{x^y} \frac{dt}{t}. If y is not 0, we can make the substitution u= t^{1/y} so that t= u^y and du= yu^{y-1}du. Also when t= 1, u= 1 and when t= x^y, u= x. ln(x^y)= \int_1^x \frac{1}{u^y}(yu^{y-1}du= y\int_1^x \frac{du}{u}= yln(x).

    If y= 0, then x^y= x^0= 1 so that ln(x^y)= ln(1)= 0= 0(ln(x))= y ln(x) so ln(x^y)= y ln(x) is true for any y.


    Since ln(x) is differentiable for all positive x, we can apply the mean value theorem on, say, the interval from x= 1 to x= 2. \frac{ln(2)- ln(1)}{2- 1}= ln(2)= \frac{1}{c} where 1\le c\le 2. Then \frac{1}{2}\le \frac{1}{c}\le 1. That is, ln(2)\ge \frac{1}{2}.

    That is important for the following reason: If X is any positive real number, then ln(2^{2X})= 2X ln(2)\ge (2X)(1/2)= X. 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: 1= \frac{1}{x}ln(y)= ln(y^{1/x}). Going back to the exponential, y^{1/x}= exp(1) so that y= (exp(1))^x. Of course, if x= 0, y= exp(0)= 1= exp(1)^0 so exp(x)= (exp(1))^x. 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 exp(x)= e^x.
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    Re: Derivative of e^x from first principles

    Quote Originally Posted by sakodo View Post
    Thanks for replying.

    Looking at posts #3 and #4 by Prove it and chisigma respectively, which one came first? The definition of e=\lim_{n\to 0} (1+\frac{1}{n})^{n} or the defintion of e^x 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.
    "Which one came first" would be relevant to an historian but is not mathematically relevant. Any equivalent definition can be used.
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    Re: Derivative of e^x from first principles

    Quote Originally Posted by HallsofIvy View Post
    define ln(x)= \int_1^x \frac{1}{t}dt. From that the fundamental theorem of calculus gives
    \frac{dln(x)}{dx}= \frac{1}{x}.
    Quote Originally Posted by chisigma View Post
    All right!... in that case I propose to all You the following 'easy little task': demonstrate on the basis of the defintion above that is...
    \ln (x_{1}\ x_{2}) = \ln x_{1} + \ln x_{2}
    Is indeed an 'easy little task'.
    Suppose that a>0~\&~b>0 then

    \ln (ab) = \int_1^{ab} {\frac{{dt}}{t}}  = \int_b^{ab} {\frac{{dt}}{t}}  + \int_1^b {\frac{{dt}}{t} = \underbrace {\int_1^a {\frac{{du}}{u}} }_{t = bu} + \int_1^b {\frac{{dt}}{t}} }  = \ln (a) + \ln (b)
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