# Thread: Is this function differentiable at zero?

1. ## Is this function differentiable at zero?

Is this function differentiable at zero?

$f(x)=\mid x\mid ^\frac{5}{2}$

I'm having a bit of trouble deciding whether functions like this are differentiable so steps would be helpful.
Thanks.

2. Originally Posted by featherbox
Is this function differentiable at zero?

$f(x)=\mid x\mid ^\frac{5}{2}$

I'm having a bit of trouble deciding whether functions like this are differentiable so steps would be helpful.
Thanks.
Since the value at 0 is 0, this function is differentiable at 0 if and only if $\frac{f(x)}{x}$ has a limit as $x\to 0$. In the case of $f(x)=|x|^\alpha$, the ratio equals $|x|^{\alpha-1}$ if $x>0$ and $-|x|^{\alpha-1}$ if $x<0$ (because then $|x|=-x$), hence
- it converges to 0 if $\alpha>1$;
- if $\alpha=1$, it converges to 1 from the right and -1 from the left, hence it doesn't converge
- if $0<\alpha<1$, it diverges to $+\infty$ from the right and $-\infty$ from the left, hence it doesn't converge.

As a conclusion, it is differentiable iff $\alpha>1$, and the derivative is 0 then.