Proving x|x| is differentiable

**monsieur fatso** · April 16th 2009, 04:31 PM

I just have a quick, and most likely silly question. I need to show that the function x|--> x|x| is differentiable on R. Would the best way to do this be to let g(x) = x, h(x) = |x|, and use the product rule to show that the derivative is 2x for all x?

Is there anything special I would need to do for x = 0, since |x| isn't differentiable at x=0?

Thanks for your help.

**skeeter** · April 16th 2009, 05:03 PM

Originally Posted by monsieur fatso

I just have a quick, and most likely silly question. I need to show that the function x|--> x|x| is differentiable on R. Would the best way to do this be to let g(x) = x, h(x) = |x|, and use the product rule to show that the derivative is 2x for all x?

Is there anything special I would need to do for x = 0, since |x| isn't differentiable at x=0?

Thanks for your help.

$f(x) = x|x|$

for the case $x = 0$ ...

$f'(0) = \lim_{x \to 0} \frac{f(x) - f(0)}{x - 0}$

$f'(0) = \lim_{x \to 0} \frac{x|x|}{x} = 0$

should be clear that the limit also exists for all $x \neq 0$

**monsieur fatso** · April 16th 2009, 05:22 PM

Okay, I guess I'm having some trouble evaluating that limit..
$f'(x) = \lim_{t\to x} \frac{f(t)-f(x)}{t-x} = \frac{x|x|-x|x|}{x-x}=|x|<br />$

Which isn't correct...
Any advice?

**skeeter** · April 16th 2009, 05:39 PM

$f'(a) = \lim_{x \to a} \frac{f(x) - f(a)}{x - a}$

$f'(a) = \lim_{x \to a} \frac{x|x| - a|a|}{x - a}$

for $x$ and $a$ both positive ...

$f'(a) = \lim_{x \to a} \frac{x^2 - a^2}{x - a} <br />$

$f'(a) = \lim_{x \to a} (x + a) = 2a$

for $x$ and $a$ both negative ...

$f'(a) = \lim_{x \to a} \frac{x(-x) - a(-a)}{x - a}$

$f'(a) = -\lim_{x \to a} \frac{x^2 - a^2}{x - a}$

$f'(a) = -\lim_{x \ to a} (x + a) = -2a<br />$

Thread: Proving x|x| is differentiable

LinkBack

Thread Tools

Display

Proving x|x| is differentiable

Similar Math Help Forum Discussions

proving that arctanh(x) is differentiable

Stewart's Proof that Inverses of Differentiable Functions are Differentiable

proving the exponential function is differentiable

Proving an equation is non-differentiable?

Sequence of differentiable functions, non-differentiable limit

Search Tags