Finding the minimum of the directional derivative of $f(x,y)=x\cos(2y)+2$ along a level curve

Question

A particle moves along the surface given by

$$f(x, y)= x \cos(2y)+2,$$

such that its coordinates are determined by the curve

$$\mathbf{r}(t)= \langle\,\ln(t +1),\sin(\pi\cdot t)\,\rangle,$$

for $0 \le t \le 2$. At the moment when $t = \frac{1}{4}$, the particle arrives at point $E$ shown on the surface.

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How do you find the minimum value of the directional derivative of $f$ at $E$?

I am having difficulty relating the $\mathbf{r}$ function with the $f$ function. Is it that $\mathbf{r}$ is just a level curve of $f$?