<s>[INST] Derivative of absolute value of $|x^5|$ 
Differentiate $|x^5|$.

I know the formula for the derivative of absolute value but I can't seem to apply it to get $5x|x^3|$.
 [/INST] Note: $x^5$ can be simplified to $x^4\cdot|x|$
Apply the product rule: $\frac{\text{d}}{\text{d}x}(x^4\cdot|x|)$:
$$\frac{\text{d}}{\text{d}x}(x^4\cdot|x|)=$$
$$\frac{\text{d}}{\text{d}x}(x^4)\cdot|x|+x^4\cdot\frac{\text{d}}{\text{d}x}(|x|)=$$
$$4x^3\cdot|x|+x^4\cdot\frac{x}{|x|}=$$
$$4x^3\cdot|x|+\frac{x^5}{|x|}=$$
Since $|x^2|=x^2$
$$4x^3\cdot|x|+\frac{x^3\cdot|x^2|}{|x|}=$$
$$4x^3\cdot|x|+x^3\cdot|x|=$$
$$5x^3\cdot|x|$$
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