Deriving the Limit Definition of |x| Using Symbolic Methods

[Orion1]

Is this the correct symbolic method to differentiate this formula using the limit definition?

$$f(x) = x|x|$$

$$f'(x) = \lim_{h \rightarrow 0} \frac{(x + h)|(x + h)| - x|x|}{h}$$

$$f'(x) = \lim_{h \rightarrow 0} \frac{(x + h)^2 - x^2}{h} = \lim_{h \rightarrow 0} \frac{x^2 + 2hx + h^2 - x^2}{h} = \lim_{h \rightarrow 0} \frac{2hx + h^2}{h} = \lim_{h \rightarrow 0} 2x + h$$
$$\boxed{f'(x) = 2|x|}$$

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[shmoe]

3 cases, x>0, x<0 and x=0.