If $u(x)$ is real and differentiable and $v(y)$ is real and differentiable, is it then given that $f(x,y) = u(x)\cdot v(y)$ is differentiable?
I have tried to make a proof in the case where $u(x) = cos(x)$ and $v(y) = sinh (y)$ from the definition of a differentiable function. But I end up with a "ugly-long" limit. So I wanted to do an easy proof where I just can deduce that $f$ is differentiable, because $u, v$ are differentiable functions.