Differential proofs

Question

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.

Answer 1

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?