Is |x|^3 differentiable?

1. Dec 5, 2007

varygoode

[SOLVED] Is |x|^3 differentiable?

1. The problem statement, all variables and given/known data

Is $$|x|^3$$ differentiable?

2. Relevant equations

$$Def: \ Let \ f \ be \ defined \ (and \ real-valued) \ on [a,b]. \ \ For \ any \ x \in [a,b], \ form \ the \ quotient$$

$$\phi(t)=\frac{f(t)-f(x)}{t-x} \ \ \ \ (a<t<b, \ t\neqx), \\$$

$$and \ define \\$$

$$f^{'}(x)=\lim_{\substack{t\rightarrow x}} \phi(t)$$

3. The attempt at a solution

Well, using the definition, I calculated that the right-hand limit and left hand limit are different. But I'm not sure if that means anything or what I can conclude here. Nor am I sure how I should define the left and right limits here.

Any help would be great, thanks!

2. Dec 5, 2007

CompuChip

If the left hand limit and the right hand limit are different, then the limit that defines the derivative at that point does not exists: the function is not differentiable.

However, I'm not sure it's not differentiable in this case... if there were a problem it would be at x = 0 and when I sketch a picture I don't immediately see a problem arising. Can you show us the calculation for the limit?

3. Dec 5, 2007

Shooting Star

The left hand derivative and the right hand derivative at x=0 are both zero. It's differentiable.

4. Dec 5, 2007

HallsofIvy

Staff Emeritus
How did you get that the right and left hand derivatives are different? If t> 0, |f(t)|= |t^3|= t^3 then at x= 0,
$$\frac{f(t)-f(x)}{t- x}= \frac{|t^3|}{t}= \frac{t^3}{t}= t^2$$
and the limit of that, as t goes to 0, is 0.

If t< 0, |f(t)|= |t^3|= -t^3. At x= 0,
$$\frac{f(t)-f(x)}{t-x}= \frac{|t^3|}{t}= \frac{-t^3}{t}= -t^2$$
but the limit, as t goes to 0, is still 0. The function is differentiable at 0 and the derivative there is 0.

Obviously, if x> 0, $f(x)=|x^3|= x^3$ which is differentiable and if x< 0, $f(x)= |x^3|= -x^3$ which is differentiable.

5. Dec 5, 2007

varygoode

Ah, I see what I did now. Totally forgot to cube my expression in my calculations, hahaha. Alrighty, thanks Ivy.

6. Dec 6, 2007

danni7070

$$|x|^3$$ the same as [tex] |x^3| [\tex] ?

7. Dec 6, 2007

Shooting Star

They are equal, not same. They are different functions.