Function $f(x,y)=u(x)v(y)$ differentiable of 2 variables

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.

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Yes, because the functions $a(x,y) = u(x)$ and $b(x,y) = v(y)$ are differentiable (as two variable functions), hence the product of $a$ and $b$, $f(x,y) = a(x,y) b(x,y)$, is differentiable.