To me: Page 126 of Cracking the GRE Math Subject Test.
A rigorous proof can be found here (Calculus, Howard Anton). A note of stress should be on “any smooth curve C on the surface”. Below is my attempt interpretation before looking at any proofs. While it cannot be considered as a proof, it shows my intuition which can be fallacious.
Let and , measures how changes as changes for which is close to . The same goes for . Now let us consider a point with approaches , we get:
Next, we consider a point with approaches , we get:
However, if is smooth at :
We then get the below formula as approaches :
So we can use the above formula to easily compute an approximation of , for is near to . The formula also represents a tangent plane.