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Math Help - Is f(x)=x|x| differentiable at x=0?

  1. #1
    Junior Member NAPA55's Avatar
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    Is f(x)=x|x| differentiable at x=0?

    I'm stuck on a homework question from my high school calculus course.

    Is f(x)=x|x| differentiable at x=0?

    I know what the graph of f(x) looks like based on Geometer's Sketchpad. I also know that a function is differentiable at a point if the derivative at that point exists.

    Does the derivative at x=0 exist? Is the function differentiable?

    Since it's an absolute value function, wouldn't the derivative be f '(x)= -2x when x<0 and f '(x)= 2x when x>0? Is there an equation for the derivative at zero?
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  2. #2
    Grand Panjandrum
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    Quote Originally Posted by NAPA55 View Post
    I'm stuck on a homework question from my high school calculus course.

    Is f(x)=x|x| differentiable at x=0?

    I know what the graph of f(x) looks like based on Geometer's Sketchpad. I also know that a function is differentiable at a point if the derivative at that point exists.

    Does the derivative at x=0 exist? Is the function differentiable?

    Since it's an absolute value function, wouldn't the derivative be f '(x)= -2x when x<0 and f '(x)= 2x when x>0? Is there an equation for the derivative at zero?
    Yes but as x \to 0 they both go to the same limit 0, so the derivative exists at x=0 and is 0.

    (more precisely the left and right derivative at x=0 both exist and are equal to the same thing: 0)

    RonL
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    Junior Member NAPA55's Avatar
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    Thanks!!
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