Problem with the Derivative Operator (')

Question

I was working through some physics equations and came to a dead stop when I couldn't get the Derivative operator to work on equations with 2 variables. An example I striped the problem down to is below. What subtle coding principal am I missing about symbolics? I have the documentation open on the other screen and it's not exactly clear how to work with pairs as inputs to functions. I looked at everything with Fullform and didn't see anything too unexpected. Is there an explanation as to what's going on?

h[h_] := h^2 + 2 h +3
h'[u]

2+2 u

(returns as expected)

f[{x_,y_}] := x^4 + y^4
Derivative[1][f][{x,y}]    
f'[{x,y}]

f′[{x,y}]

f′[{x,y}]

Answer 1

When a function has 2 arguments (not a single list argument), use:

f[x_, y_] := x^4 + y^4
Derivative[1, 0][f][x, y]
Derivative[0, 1][f][x, y]

4 x^3

4 y^3

Answer 2

Here are two approaches. The first requires perhaps more tolerance for "noisy" notation. Note that I did not use a vector argument. If you must, the notation will be correspondingly "noisier".

Clear[f, x, y]
Dt[f[x, y]]
Grad[f[x, y], {x, y}].{dx, dy}