Chain Rule

In[146]:=

  Clear[f,x,y,t]
  D[f[x[t],y[t]],t]

Out[146]=

         (0,1)                      (1,0)
  y'[t] f     [x[t], y[t]] + x'[t] f     [x[t], y[t]]

Let's look at it in another way.

In[147]:=

  Clear[f]
  grad2[f][x[t],y[t]].{x'[t],y'[t]}

Out[147]=

         (0,1)                      (1,0)
  y'[t] f     [x[t], y[t]] + x'[t] f     [x[t], y[t]]

If you have a parametric curve r(t)=(x(t), y(t)), the dot (inner) product of the gradient vector with the derivative vector of the curve is the derivative of f(r(t)) or f(x(t),y(t)).

Up to The Gradient