The derivative of f(x) = |x|^3 at x = 0 is | vedclass

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(A) Given the function $f(x) = |x|^3$. We can write this as $f(x) = x^3$ for $x \ge 0$ and $f(x) = -x^3$ for $x < 0$. To find the derivative at $x = 0

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(A) Given the function $f(x) = |x|^3$. We can write this as $f(x) = x^3$ for $x \ge 0$ and $f(x) = -x^3$ for $x To find the derivative at $x = 0$,we calculate the left-hand derivative $(LHD)$ and right-hand derivative $(RHD)$. $LHD = \lim_{h 	o 0^-} \frac{f(0+h) - f(0)}{h} = \lim_{h 	o 0^-} \frac{|h|^3 - 0}{h} = \lim_{h 	o 0^-} \frac{-h^3}{h} = \lim_{h 	o 0^-} (-h^2) = 0$. $RHD = \lim_{h 	o 0^+} \frac{f(0+h) - f(0)}{h} = \lim_{h 	o 0^+} \frac{|h|^3 - 0}{h} = \lim_{h 	o 0^+} \frac{h^3}{h} = \lim_{h 	o 0^+} (h^2) = 0$. Since $LHD = RHD = 0$,the derivative $f'(0)$ exists and is equal to $0$.

The derivative of $f(x) = |x|^3$ at $x = 0$ is

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