Difference between 2 types of differentials?

[EV33]

I have a simple question about differentials. I have been taught two ways to find the differential and my questions is in what situations do I use each one?

simply speaking these are the 2 ways

1.) just take the partials of each component function and throw them in a matrix

2.) Let f be the function we want the differential for, where f:S→ℝ^m. You choose a curve $\alpha$(-ε,ε)→S such that $\alpha$(0)=p and $\alpha$'(0)= v where p is in S and S is a surface, and v is in TpS.
Then you compose f with $\alpha$ and take the derivative with respect to t.

Thank you for your time.

 

[HallsofIvy]

In 1, it looks like you are referring to the differential of a function from $R^m$ to $R^n$ while in 2, you are specifically referring to a function from $R^1$ to $R^n$.

Also, 1 is assuming a specific coordinate system while 2 is a general, non-coordinate definition.