Which of the following functions is different | vedclass

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(D) Let $f(x) = \sin(|x|) - |x|$. We check the differentiability at $x = 0$ by calculating the Left Hand Derivative $(LHD)$ and Right Hand Derivative

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$\sin (|x|) - |x|$

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(D) Let $f(x) = \sin(|x|) - |x|$. We check the differentiability at $x = 0$ by calculating the Left Hand Derivative $(LHD)$ and Right Hand Derivative $(RHD)$.For $x For $x > 0$,$|x| = x$,so $f(x) = \sin x - x$. The derivative is $f'(x) = \cos x - 1$. Thus,$RHD = \lim_{x 	o 0^+} (\cos x - 1) = \cos(0) - 1 = 1 - 1 = 0$.Since $LHD = RHD = 0$,the function $\sin(|x|) - |x|$ is differentiable at $x = 0$.For the other options,$\cos(|x|)$ is differentiable at $x=0$ but $|x|$ is not,so their sum or difference is not differentiable at $x=0$. Similarly,$\sin(|x|) + |x|$ has $LHD = -2$ and $RHD = 2$,so it is not differentiable.

Which of the following functions is differentiable at $x = 0$?

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