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Math Help - proofs for the derivatives of logax, lnx, a^x and e^x

  1. #1
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    proofs for the derivatives of logax, lnx, a^x and e^x



    I need short, sweet, simple proofs for all the derivatives of log {a} {x}, ln{x}, a^x and e^x concurrently so as to avoid a "chicken or the egg" situation in which 1 proof relies on one of the other 3 results.

    Note:
    {d(\log{a}{x}})/{dx} = 1/{x\ln{a}}
    d(\ln{x})/{dx} = 1/x
    d(a^x)/dx = a^x\ln{a}
    d(e^x)/dx = e^x

    Caution: Do not commit circular reasoning like what I have encountered.

    Anyone who can do so is likely to benefit most of us who are learning derivatives.

    "Proof is what sets mathematics apart from every other science." ----Prove It, MHF Contributor

    Thank You.
    Last edited by normalguy; April 10th 2011 at 07:16 AM.
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  2. #2
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    Hi normalguy,

    It all depends on how you define those functions.

    So, what is your definition of \ln x? How about e^x?
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  3. #3
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    Quote Originally Posted by awkward View Post
    Hi normalguy,

    It all depends on how you define those functions.

    So, what is your definition of \ln x? How about e^x?
    ????
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  4. #4
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    Quote Originally Posted by normalguy View Post
    ????
    definition of the transcendental number e ...

    \displaystyle e = \lim_{n \to \infty} \left(1 + \frac{1}{n}\right)^n

    or

    \displaystyle e = \lim_{x \to 0} \left(1 + x\right)^{\frac{1}{x}}


    definition of the natural log function ...

    \displaystyle \ln{x} = \int_1^x \frac{1}{t} \, dt



    you have to start with one or the other.
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  5. #5
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    Start by defining \displaystyle e^x as the function that is its own derivative (for that IS the definition of \displaystyle e^x).

    The other proofs follow with some algebraic manipulation.
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  6. #6
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    Quote Originally Posted by skeeter View Post
    definition of the transcendental number e ...

    \displaystyle e = \lim_{n \to \infty} \left(1 + \frac{1}{n}\right)^n

    or

    \displaystyle e = \lim_{x \to 0} \left(1 + x\right)^{\frac{1}{x}}


    definition of the natural log function ...

    \displaystyle \ln{x} = \int_1^x \frac{1}{t} \, dt



    you have to start with one or the other.
    can you explain further on this? why is e=(1+y)^y
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  7. #7
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    so is the proof of the derivative of logax dependent on the result of the derivative of lnx or vice versa?
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  8. #8
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    Quote Originally Posted by normalguy View Post
    so is the proof of the derivative of logax dependent on the result of the derivative of lnx or vice versa?
    if you accept the fact that \dfrac{d}{dx} (\ln{x}) = \dfrac{1}{x} , then ...

    \dfrac{d}{dx} (\ln{ax}) = \dfrac{d}{dx} (\ln{a} + \ln{x}) = 0 + \dfrac{1}{x} = \dfrac{1}{x}




    Quote Originally Posted by normalguy View Post
    can you explain further on this? why is e=(1+y)^y
    that equation is not what I posted ... if you want what you call "a non-circular proof", then you need the necessary background information to understand the proof.

    e (mathematical constant) - Wikipedia, the free encyclopedia
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  9. #9
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    I expect the OP meant \displaystyle \log_a{x}, not \displaystyle \log{ax}.

    Notice that any logarithm can be changed to the natural logarithm...

    \displaystyle y = \log_a{x}

    \displaystyle a^y = x

    \displaystyle \ln{\left(a^y\right)} = \ln{x}

    \displaystyle y\ln{a} = \ln{x}

    \displaystyle y = \frac{\ln{x}}{\ln{a}}.

    So what is the derivative of \displaystyle \log_a{x}?
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  10. #10
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    Quote Originally Posted by Prove It View Post
    I expect the OP meant \displaystyle \log_a{x}, not \displaystyle \log{ax}.

    Notice that any logarithm can be changed to the natural logarithm...

    \displaystyle y = \log_a{x}

    \displaystyle a^y = x

    \displaystyle \ln{\left(a^y\right)} = \ln{x}

    \displaystyle y\ln{a} = \ln{x}

    \displaystyle y = \frac{\ln{x}}{\ln{a}}.

    So what is the derivative of \displaystyle \log_a{x}?
    I got it. 1/xlna

    so you are using d/dx(lnx)=1/x to proof the derivative of \displaystyle \log_a{x}

    But I am curious to know which was proven first? derivative of \displaystyle \log_a{x} or  ln{x}?

    My textbook says in the special case where a=e for \displaystyle y = \log_a{x}, the derivative is 1/xlne=1/x. Hence I assume derivative of log_a{x} should be proven first!

    But your working seems to suggest otherwise.
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  11. #11
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    meanwhile, some1 explain this thingy \displaystyle e = \lim_{n \to \infty} \left(1 + \frac{1}{n}\right)^n which apparently seems to be the basis of all the proofs, right?
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  12. #12
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    Quote Originally Posted by normalguy View Post
    I got it. 1/xlna.
    That is correct is read correctly.
    You really do need to learn to use symbols.
    Why not learn to post in symbols? You can use LaTeX tags
    [tex]y=\log_a(x)~\Rightarrow y'=\dfrac{1}{x\ln(a)}[/tex] gives y=\log_a(x)~\Rightarrow y'=\dfrac{1}{x\ln(a)}

    Not being able to read your post slows us down.
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  13. #13
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    Quote Originally Posted by normalguy View Post
    meanwhile, some1 explain this thingy \displaystyle e = \lim_{n \to \infty} \left(1 + \frac{1}{n}\right)^n which apparently seems to be the basis of all the proofs, right?
    There are several ways to approach this question. They start at different places and end up at that conclusion.
    Defining \mathbf{e} as the number such that \int_1^e {\frac{{dx}}{x}}  = 1 ties \mathbf{e}^x and \ln(x) together.

    The using the monotone property of integrals we can get \displaystyle e^{\frac{n}{{n + 1}}}  \leqslant \left( {1 + \frac{1}{n}} \right)^n  \leqslant e.

    The limit you asked about is the next step.
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  14. #14
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    Hello, normalguy!

    Do you really want four separate proofs?
    That is a lot of unnecessary work.
    If you insist on four proofs, I suggest you find them yourself.


    \text{I need short/sweet/simple proofs for the derivatives of }\log_a x,\;\ln x,\;a^x
    \text{ and }e^x\,\text{ concurrently so as to avoid a "chicken or the egg" situation}
    \text{in which one proof relies on one of the other 3 results.}

    . . \dfrac{d}{dx}(\log_ax) \:=\: \dfrac{1}{x\ln a}

    . . \dfrac{d}{dx}(\ln x ) \:=\: \dfrac{1}{x}

    . . \dfrac{d}{dx}(a^x) \:=\: a^x\ln{a}

    . . \dfrac{d}{dx}(e^x) \:=\: e^x
    .

    I'll prove the first one . . . The others will follow nicely.


    We have: . f(x) \;=\;\log_a(x)

    f(x+h) - f(x) \;=\;\log_a(x+h) - \log_a(x) \;=\;\log_a\left(\dfrac{x+h}{x}\right)


    \displaystyle \frac{f(x+h)-f(x)}{h} \;=\;\frac{1}{h}\log_a\left(\frac{x+h}{x}\right) \;=\; \log_a\left(1 + \frac{h}{x}\right)^{\frac{1}{h}}

    . . . . . . . . . . . . . \displaystyle =\;\log_a\left[\left(1 + \frac{h}{x}\right)^{\frac{x}{h}}\right]^{\frac{1}{x}}  \;=\;\frac{\ln\left[\left(1+\frac{h}{x}\right)^{\frac{x}{h}}\right]^{\frac{1}{x}}}{\ln a}


    \displaystyle f'(x) \;=\;\lim_{h\to0}\left\{ \frac{\ln\left[\left(1+\frac{h}{x}\right)^{\frac{x}{h}}\right]^{\frac{1}{x}}}{\ln a}\right\} \;=\;\frac{\ln\overbrace{\left[\lim\left(1 + \frac{h}{x}\right)^{\frac{x}{h}}\right]}^{\text{This is }e}^{\frac{1}{x}}}{\ln a}

    . . . . . \displaystyle =\;\frac{\ln(e^{\frac{1}{x}})}{\ln a} \;=\; \frac{\frac{1}{x}}{\ln a}


    \displaystyle \text{Therefore: }\;\frac{dx}{dx}(\log_a x) \;=\;\frac{1}{x\ln a}

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  15. #15
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    i manage to figure out the proofs of derivatives in the following order:
    1. ln(x)<br />
2. \log_a x<br />
3. e^x<br />
4. a^x<br />
    am i right to say that the basis of all proofs is the definition of e?
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