Differential of a direct product

Question

Given two differentiable maps between differential manifolds:

$$f : X \to Y, g : X \to Z,$$

Is $f \times g : X \times X \to Y \times Z$ differentiable?

I think yes. And you get $d(f \times g)_{(x,y)} = d(f)_x \times d(g)_y$?