This is outraging. Help with multivariable calculus!!!!!!!!!!!!?

Question

People that explain things on the web don't explain you why rules exist in mathamatics. I don't like to be a puppet, I like to know what I'm doing, and to do so, I must know "why" I'm doing.
Can someone brilliant explain me why the 2nd derivative test of multivariable calculus is so weird??? To help, I've seen it named D in many online courses. (It is the difference between the product of the 2nd order partial derivatives with respect to x and to y times the square of the partial derivative with respect to x then y).

In the past I've managed to figure out the "why"s and the "what"s on my own, but this time its too complicated. Damn those teachers who think that all students only need to get a good grade!!!!!!!!!

Answer 1

The D test in multivariable calculus is very similar to the one variable test for concavity. The reason the D test is more complicated is because graphs in 3 dimensions can curve differently along separate dimensions. For instance, if you were to take the second partial derivative with respect to x of the function, and the result was negative, that would mean that, in the yz plane that cuts through a particular "critical point", the projection of the graph into that plane is concave down. If you were to take the second partial derivative with respect to y and the result were positive, that would mean that projection of the graph into the xz plane through a critical point is concave up. Try visualizing different situations in which you have concave up in one dimension and concave down in another direction and then you will understand the D test a little better.

Answer 2

The D is often used with total derivative such that

Df/Ds = df/dx * dx/ds + df/dy * dy/ds

to get the second derivative you do the same again:

D/Ds(Df/Ds) = ....

Remember that Df or df are finite differences until taken to the limit when ds goes to zero.

ASCII really isn't appropriate for this.