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Math Help - Derivative Functions - Is This a Proper Proof?

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    Derivative Functions - Is This a Proper Proof?

    I know that f(x) is derivative at x=0, and I need to prove the following, while x<a<y :

    Is this a proper proof? Thank you very much!




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  2. #2
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    Quote Originally Posted by adam63 View Post
    I know that f(x) is derivative at x=0, and I need to prove the following, while x<a<y :

    Is this a proper proof? Thank you very much!




    What is this? What is f(x)? There're lots of functions which are derivable at some point but not at another point...

    Tonio
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    Quote Originally Posted by tonio View Post
    What is this? What is f(x)? There're lots of functions which are derivable at some point but not at another point...

    Tonio
    I need to prove it the way it is :
    let f(x) be derivative at x=a, and I need to prove that:
    lim\frac{f(x)-f(y)}{x-y} = f'(a)
    (while x,y \rightarrow a ; x<a<y )
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  4. #4
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    Look at \frac{f(x)- f(y)}{x- y}= \frac{f(x)- f(a)}{x- y}- \frac{f(y)- f(a)}{x- y} =\left(\frac{x-a}{x- y}\right)\frac{f(x)- f(a)}{x- a}-\left(\frac{y-a}{x-y}\right)\frac{f(y)- f(a)}{y- a}
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    Quote Originally Posted by HallsofIvy View Post
    Look at \frac{f(x)- f(y)}{x- y}= \frac{f(x)- f(a)}{x- y}- \frac{f(y)- f(a)}{x- y} =\left(\frac{x-a}{x- y}\right)\frac{f(x)- f(a)}{x- a}-\left(\frac{y-a}{x-y}\right)\frac{f(y)- f(a)}{y- a}
    Thank you for your suggestion!

    Is my way wrong? If so, where is it wrong (I'm a 'calculus beginner', so I will be glad to know where I should fix myself)

    Thank you
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    And by the way, I didn't understand how to continue your suggestion for a proof, Tonio. Where does the part of x,y \rightarrow a appears?

    Thank you very much!
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  7. #7
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    Quote Originally Posted by adam63 View Post
    Thank you for your suggestion!

    Is my way wrong? If so, where is it wrong (I'm a 'calculus beginner', so I will be glad to know where I should fix myself)

    Thank you
    The problem is that you don't show how to go from one step to the next. The next to last line is \lim_{x,y\to a}\frac{f(y)- f(x)}{y- x} and the line just before that is \lim_{x,y\to a}\frac{f(x+ y- a)- f(x)}{y- x}. How did you get from one to the other?
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  8. #8
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    That's because:

    x \rightarrow a, then (x-a) \rightarrow 0.
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