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The absolute value function is defined as: $$|x| = \begin{cases} x, & \text{if } x > 0 \\ -x, & \text{if } x < 0 \end{cases}$$ Note that the function is not differentiable at $x=0$.
If $x > 0$, then $|x| = x$. Therefore, $$\frac{d}{dx}(|x|) = \frac{d}{dx}(x) = 1$$ Also, if $x > 0$, then $\frac{x}{|x|} = \frac{x}{x} = 1$. Thus, for $x > 0$, $\frac{d}{dx}(|x|) = \frac{x}{|x|}$.
If $x < 0$, then $|x| = -x$. Therefore, $$\frac{d}{dx}(|x|) = \frac{d}{dx}(-x) = -1$$ Also, if $x < 0$, then $\frac{x}{|x|} = \frac{x}{-x} = -1$. Thus, for $x < 0$, $\frac{d}{dx}(|x|) = \frac{x}{|x|}$.
Since $\frac{d}{dx}(|x|) = \frac{x}{|x|}$ for both $x > 0$ and $x < 0$, we can conclude that $$\frac{d}{dx}(|x|) = \frac{x}{|x|}, \quad x \ne 0$$
Final Answer: $\frac{d}{dx}(|x|)=\frac{x}{|x|},x\ne0$
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