Question

is f(x) = ⎜x+∏⎟sin(x) differentiable on R?

Answer 1

Yes it is. Since the sin-function is differentiable everywhere and the absolute value is differentiable everywhere except at the origin. Here sin(x) goes to zero at -π where the absolute value would be non-differentiable.

Answer 2

To clarify, both the absolute value and the sin(x) goes to zero at -π, which makes it smooth there because of the smooth nature of sin.