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Math Help - Is this correct

  1. #1
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    Is this correct

    Suppose that the function 0, \infty) \rightarrow R" alt="f0, \infty) \rightarrow R" /> is differentiable and let c>0. Now define 0, \infty) \rightarrow R " alt="g0, \infty) \rightarrow R " /> by g(x) = f(cx) for x>0. Just using the definition of derivative, show that g'(x) = c f'(x) for x>0

    My work:

    according to the defnition of derivative, g(x) is diffrentiable if the following limit exists

    \lim_{x \to {x_0}} \frac{g(x)-g(x_0)}{x-{x_0}}

    \lim_{x \to {x_0}} \frac{f(cx)-f(cx_0)}{x-{x_0}}

    c \lim_{x \to {x_0}} \frac{f(cx)-f(cx_0)}{c(x-{x_0})}

    c \times f'(cx)
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  2. #2
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    Quote Originally Posted by serious331 View Post
    Suppose that the function 0, \infty) \rightarrow R" alt="f0, \infty) \rightarrow R" /> is differentiable and let c>0. Now define 0, \infty) \rightarrow R " alt="g0, \infty) \rightarrow R " /> by g(x) = f(cx) for x>0. Just using the definition of derivative, show that g'(x) = c f'(x) for x>0

    My work:

    according to the defnition of derivative, g(x) is diffrentiable if the following limit exists

    \lim_{x \to {x_0}} \frac{g(x)-g(x_0)}{x-{x_0}}

    \lim_{x \to {x_0}} \frac{f(cx)-f(cx_0)}{x-{x_0}}

    c \lim_{x \to {x_0}} \frac{f(cx)-f(cx_0)}{c(x-{x_0})}

    c \times f'(cx)

    Very nice. One step before the last one though, I'd rather write:

    \lim_{cx\to cx_0}\,c\,\frac{f(cx)-f(cx_0)}{cx-cx_0}

    Tonio
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  3. #3
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    Quote Originally Posted by tonio View Post
    Very nice. One step before the last one though, I'd rather write:

    \lim_{cx\to cx_0}\,c\,\frac{f(cx)-f(cx_0)}{cx-cx_0}

    Tonio
    Hi tonio,

    I have actually written that in my notebook, but was too lazy to write that line in this thread. Thanks for your time!
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