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Math Help - Determining if a function is differentiable

  1. #1
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    Determining if a function is differentiable

    I can't figure out how to do this at all. Basically...the GRAPH looks differentiable, and I know from my book that g(x) = (x + |x|)^2 + 1 is differentiable, so...shouldn't this one be too? According to my math (taking the limit of both sides)...it's not. I get the left hand limit is -1 and the right hand limit is 0. And I don't get why, or if that's right...?






    Problem: Determine whether or not the function g(x) = (x + |x|)^2 - 1 is differentiable at x = 0. If so, find g\prime(0).

    Also, to do it, I can only use a difference quotient (I haven't learned any fancy shortcuts yet):

    f\prime(x) = \frac{f(x + h) - f(x)}{h}
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  2. #2
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    to check differentiability, the following limit must exist \underset{x\to 0}{\mathop{\lim }}\,\frac{g(x)-g(0)}{x-0}, and not only that, check the one-sided limits, they must exist and be equal each other.
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  3. #3
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    Quote Originally Posted by Krizalid View Post
    to check differentiability, the following limit must exist \underset{x\to 0}{\mathop{\lim }}\,\frac{g(x)-g(0)}{x-0}, and not only that, check the one-sided limits, they must exist and be equal each other.
    Also you might notice that: g(x) = \left\{ {\begin{array}{rl}<br />
   {4x^2  - 1,} & {0 \leqslant x}  \\<br />
   { - 1,} & {x < 0}  \\<br /> <br />
 \end{array} } \right.
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