Differential proofs We had an assignment for the past week, to prove some equations which I had no idea that could actually be proved, since I was kind of taking them for granted. The professor did not check them anyway he just moved on. These are : 


*

*$d(x \pm y) = dx \pm dy$

*$\Delta(x \pm y) = \Delta x \pm \Delta y$

*$x \pm y = \int dx \pm \int dy$

*$d(xy) = x\, dy + y\, dx$

*$d(x/y) = (y\, dx - x\, dy)/y^2$


How can these be proved? Thanks in advance.
 A: Strictly speaking, the operations of classical (as opposed to non-standard) calculus act on functions (which have a formal, set-theoretic definition), not on variables. Classically, however, mathematical notation indicates functional relationships by using variables as templates, e.g., writing $y = x^{2}$ to indicate the relationship of squaring, leaving the reader to interpret $x$ as an arbitrary input (or "element of the domain") and $y$ as the output.
Assuming you're in a beginning calculus course (probably for engineers or scientists...?), I'd recommend interpreting your formulas as follows, for arbitrary differentiable functions $f$ and $g$:


*

*$(f \pm g)' = f' \pm g'$.

*$(f \pm g)(x + h) - (f \pm g)(x) = \bigl[f(x + h) - f(x)\bigr] \pm \bigl[g(x + h) - g(x)\bigr]$.

*$f \pm g = \int f' \pm \int g'$ (in which the integral sign signifies taking an antiderivative).

*$(fg)' = fg' + f'g$.

*$(f/g)' = (gf' - fg')/f^{2}$.


The question becomes: Do you know how to prove each of these?
