Understanding Derivatives of f(r) in Multiple Variables

[Jhenrique]

I have some important questions and essentials for understand some theories. They are six:
given f(r(t)), f(r(u, v)), f(r(u(t), v(t))), f(r(t)), f(r(u, v)) and f(r(u(t), v(t))). How compute its derivatives wrt independent variables?

Unfortunately, I just know the answer for 1nd:
$$\frac{df}{dt}=\frac{df}{d\vec{r}}\cdot \frac{d\vec{r}}{dt}$$
I don't know equate the other derivatives.

 

[Simon Bridge]

Where you have more than one variable, you use partial derivatives.
When the variables are nested, just repeat the chain rule.
Recall how to differentiate a vector.

 

[Jhenrique]

But I think that chain rule will expand to much. I think if exist another way more compact for these differentations... some so so like
$\frac{d^2f}{du dv}=\frac{d^2 f}{d\vec{r}^T d\vec{r}}\cdot \frac{d\vec{r}}{du}\times \frac{d\vec{r}}{dv}$