Is |x|^3 Differentiable at x=0?

[varygoode]

[SOLVED] Is |x|^3 differentiable?

Homework Statement

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

Homework Equations

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

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

$$and \ define \\$$

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

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!

 

[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?

 

[Shooting Star]

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

 

[HallsofIvy]

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.

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.

 

[varygoode]

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

 

[danni7070]

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

 

[Shooting Star]

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