Question

Write the derivative of f (x) = |x|³ at x = 0.

Answer 1

Given: $f\left(x\right)={\left|x\right|}^3$ $=\left\{\begin{array}{l}{x}^3,x\ge 0\\ -x^3,x<0\end{array}\right.$

(LHD at x = 0)
$$\begin{gathered} \underset{x\to 0^{-}}{\lim }\frac{f\left(x\right)-f\left(0\right)}{x-0} \\ =\underset{h\to 0}{\lim }\frac{f(0-h)-f\left(0\right)}{x} \\ =\underset{h\to 0}{\lim }\frac{h^3}{-h} \\ =0 \end{gathered}$$.

(RHD at x = 0)

$$\begin{gathered} \underset{x\to 0^{+}}{\lim }\frac{f\left(x\right)-f\left(0\right)}{x-0} \\ =\underset{x\to 0^{+}}{\lim }\frac{f(0+h)-f\left(0\right)}{x} \\ =\underset{h\to 0}{\lim }\frac{h^3-0}{h} \\ =0 \end{gathered}$$

and $f\left(0\right)=0.$

Thus, (LHD at x=0) = (RHD at x = 0) = $f\left(0\right)$

Hence, $\underset{x\to 0}{\lim }\frac{f\left(x\right)-f\left(0\right)}{x-0}=f\text{'}\left(0\right)=0$