# Math Help - check d differentiability

1. ## check d differentiability

will d following function be differentiable at x = 0 ???
1. cos(|x|)+|x|
2. sin(|x|)-|x|

2. Hi

First one:

If $h$ approaches zero from the right, we have:

$\frac{f(x+h)-f(x)}{h} = \frac{cos(h)+h-1}{h}=\frac{cos(h)-1}{h}+\frac{h}{h} \to 1 \mbox{ as } h \to 0$

Now we approach from the left. Note that $cos(h)=cos(-h)$ , $cos(x)$ is an even function.

$\frac{cos(h)-h-1}{h}=\frac{cos(h)-1}{h}-\frac{h}{h} \to -1 \mbox{ as } h \to 0$

So the function is not differentiable at $x = 0$ .

The second one you basically do the same thing.

$\frac{sin(h)-h}{h} \to 1-1 = 0$ and

$\frac{-sin(h)+h}{h} \to -1 +1 = 0$ .

So the second one is differentiable.