That is such a lovely looking sum where did you did it? there are tones http://www.freebookcentre.net/maths-books-download/Introduction-to-Modern-Set-Theory.html and on  the internet, and cosmolearn and on youtube might help you. here is a good book on single variable calculus file:///home/chronos/u-2446e8898a70ab351e57606f4edc35950b54a39b/Downloads/Single_Variable_Calculus.pdf, mit sight might have a few materials.

I am at ap calculus level now, and I am wondering wethere I should finish doing calculus on khan academy or read and study group, field, set and chaos theory the seperet books in the basic maths section on the first hyperlink I gave you as prerequisite for abstract algebra when the time comes and some other prerequisite, or just finish calculus on khan academy then read the group, set, chaos books.

I think Graph and number theory and the constant sequence books on the first website I sent would be good for learning calculus because they do alot of graphs, sequence and constants in calculus.

Just looking at the partial derivative with respect to x, I believe the chain rule is being applied to Q, and then to R along with the product rule. The chain rule is being used because z is itself a function of x and y. The partial derivative of x with respect to x is 1, the partial of y with respect to x is 0, and the partial derivative of z with respect to x exists because z is a function of x and y. That's why it is the partial of Q with respect to x and the partial of Q with respect to z times the partial of z with respect to x. The partial of x w/ respect to x and the partial of y w/ respect to x just don't appear because it is simplified.

Is the formula in the above picture verified to be correct by a source, such as your teacher?