To prove a function f(x) is differentiable in the domain of reals? Watch
Claim: f(x)=sin(2x) is differentiable on R. Let f(x)=g(h(x)) where g(x)=sin(x) and h(x)=2x. Indeed g(x) is differentiable on R and moreover g'(x)=cos(x) also h(x)=2x is differentiable on R and moreover h'(x)=2 for all x in R.
So by the chain rule f(x)=g(h(x))=g o h is differentiable on R and moreover f'(x)=2cos(2x).
I hope that makes sense.
Obviously there may be a case where f is differentiable on all of R except some points in which case you may need to bring out the definition to see what happens at those points.
Can you just differentiate it using normal rules (product, sum, chain etc) and show the derivative you get exists for all values of x out of reals?
But even if a function is, the statement remains false - a standard example is . Differentiable everywhere, yet the derivative will appear to suggest otherwise.