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Differentiability — MATHS STD 12 Science — Question

Rajasthan BoardEnglish MediumSTD 12 ScienceMATHSDifferentiability3 Marks
Question
Write the derivative of $f(x) = |x|^3$ at $x = 0.$
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Answer

Given: $\text{f(x)}=|\text{x}^3|=\begin{cases}\text{x}^3,&\text{x}\geq0\\-\text{x}^3,&\text{x}<0\end{cases}$
$(\text{LHL}$ at $x = 0)$
$\lim_\limits{\text{x}\rightarrow0^{-}}\frac{\text{f(x)}-\text{f}(0)}{\text{x}-0}$
$=\lim_\limits{\text{x}\rightarrow0}\frac{\text{f}(0-\text{h})-\text{f}(0)}{\text{x}}$
$=\lim_\limits{\text{x}\rightarrow0}\frac{\text{h}^3}{-\text{h}}$
$=0$
$(\text{RHL}$ at $x = 0)$
$\lim_\limits{\text{x}\rightarrow0^{+}}\frac{\text{f(x)}-\text{f}(0)}{\text{x}-0}$
$=\lim_\limits{\text{x}\rightarrow0}\frac{\text{f}(0+\text{h})-\text{f}(0)}{\text{x}}$
$=\lim_\limits{\text{x}\rightarrow0}\frac{\text{h}^3-0}{-\text{h}}$
$=0$
And $f(0) = 0.$
Thus, $(\text{LHL}$ at $x = 0) = (\text{RHL}$ at $x = 0) = f(0)$
Hence, $\lim_\limits{\text{x}\rightarrow0}\frac{\text{f(x)}-\text{f}(0)}{\text{x}-0}=\text{f}\ '(0)=0$.

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