Find all values of x for which the function is differentiable.
$\displaystyle P(x)= sin(|x|)-1$
To shorten the below: no.
If something is not differentiable only when it at the very middle (where the right and left come together, usually in examples as (0,0,) of cusps, corners, vertical tangents, and a non-removable discontinuity, then I cannot think or see, using a graphing calculator, any points.
$\displaystyle f(x) = |x|$
Consider x near 0. When x approaches 0 from the left, the first derivative is -1. When x approaches 0 from the right, the first derivative is 1. So the derivative of |x| does not exist at x = 0.
We can make a similar argument for $\displaystyle f(x) = sin(|x|) - 1$. See the graph below. Notice the cusp at x = 0.
-Dan