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Can we find the derivative of $|x^3|$ at $x = 0$?

I was trying to find the derivative of $|x^3|$ within the range of $[-1/2, 1/2]$.

I got the equation for the derivative of $|x^3| = 3x^3 / |x|$. for $x \neq 0$.

Is the equation correct?

And the other thing is I want the derivative of $|x^3|$ at $x = 0$ to obtain the full solution. How can I find it?

Any help would be appreciated.

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3 Answers

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If $f(x)=|x^3|$, then by the definition of the derivative,$$ f'(0)=\lim_{x\to0}\frac{f(x)-f(0)}{x-0}=\lim_{x\to 0}\frac{|x^3|}{x} \, . $$Since$$ \lim_{x\to0^+}\frac{|x^3|}{x}=\lim_{x\to0^+}\frac{x^3}{x}=\lim_{x\to 0^+}x^2=0 \, , $$and$$ \lim_{x\to 0^-}\frac{|x^3|}{x}=\lim_{x\to0^-}\frac{-x^3}{x}=\lim_{x\to0^-}-x^2=0 \, , $$we find that that $f'(0)=0$. Alternatively, we could use the following theorem:

Suppose that $f$ is an even function, and $f'_+(x)$ exists and equals $c$. Then, $f'_-(-x)=-c$.

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This is really very simple.

If $x\ge0$, then $f(x)=x^3$ has derivative $3x^2$; so the right derivative at $x=0$ is $0$.

If $x\le0$, then $f(x)=-x^3$ has derivative $-3x^2$; so the left derivative at $x=0$ is $0$.

So the left derivative is equal to the right derivative, and therefore the derivative is their common value, $0$.

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As an alternative, following your first idea, we can use that for $x\neq 0$

$$(|x|)'=\operatorname{sign}(x)$$

then by chain rule and since $x\operatorname{sign}(x)=|x|$

$$f(x)=|x^3|=x^2|x| \implies f'(x)=(|x^3|)'=(x^2|x|)'=2x|x|+x^2\operatorname{sign}(x) =3x|x|=\frac{3x^3}{|x|}$$

and therefore by this theorem

$$\lim_{x\to 0^+}f'(x)=\lim_{x\to 0^-}f'(x) =0 \implies f'(0)=0$$

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