Question

Why do we always take the partial derivative of a surface, rather than the 'whole' derivative?

Answer 1

Would appreciate if you could provide examples

Answer 2

consider the surface \[z = x^2 + xy + y^2\]

Answer 3

For surfaces that are in 3 dimensional space, y may be a function of x. So in calculating dz/dx for examples, we will incur a "dy/dx", which is rather a pain in the neck. Partial derivatives allow us to take it wrt one variable, treating the other as a constant, which means we are looking at one dimensional space, which is much much easier