Browse Questions

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

$\begin{array}{1 1}(a)\;\cos(|x|)+|x|&(b)\;\cos(|x|)-|x|\\(c)\;\sin(|x|)+|x|&(d)\;\sin(|x|)-|x|\end{array}$

Let $f(x)=\sin(|x|)-|x|$
$Lf'(0)=\lim\limits_{h\to 0}\large\frac{f(0-h)-f(0)}{-h}$
$\qquad\;\;=\lim\limits_{h\to 0}\large\frac{\sin h-h-0}{-h}\qquad$$h > 0 \qquad\;\;=\lim\limits_{h\to 0}\large\frac{\sin h}{-h}$$+1=-1+1=0$
Hence (d) is the correct answer.