How do you calculate the directional derivative in multiple dimensions?

I've seen lots of examples of calculating the directional derivative where the solution takes the total derivative and then calculates the dot product with the unit directional vector. However this can't make sense when computing the directional derivative for a function mapping from $\mathbb{R^2} \rightarrow \mathbb{R^3}$ like in this example...

$$u = (1,2)/\sqrt{5} \ \ \ \ \ \ \ \ \ f(x,y) = (x^2y,sin(x+y),e^{xy})$$ .

Since I thought the total derivative would be:

$$\Delta f = \begin{pmatrix} 2xy & x^2\\ cos(x+y) & cos(x+y) \\ ye^{xy} & x e^{xy} \end{pmatrix}$$

...however it would be impossible to calculate the dot product with $u$. So how do you take the directional derivative in this case, and does the dot product intuition still make sense in this case (i.e. are we still looking at the component of each partial derivative in the $u$ direction)?

• Only scalar-valued functions have directional derivatives that can be interpreted without resorting to "abstract multivariate differential calculus". – Christian Blatter Feb 7 '15 at 19:30