# Thread: Is f(x)=x|x| differentiable at x=0?

1. ## 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?

2. Originally Posted by NAPA55
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

3. Thanks!!