Proving Rules of Differentials

Question

Show that for any two functions $f(x,y)$ and $g(x,y)$ we have $d(f+g)=df+dg$.

Not too sure how to go about this. How would I set up the question?

Answer 1

HINT: Use this fact: $$df=\frac{\partial f}{\partial x}dx+\frac{\partial f}{\partial y}dy$$

Answer 2

The differential operator $d/dx$ is defined as:

$$\dfrac{d f(x)}{dx}=\lim_\limits{h\to0} \dfrac{f(x+h)-f(x)}{h}$$

So $d(f+g)$ becomes:

$$\dfrac{d [f(x)+g(x)]}{dx}=\lim_\limits{h\to0} \dfrac{f(x+h)+g(x+h)-f(x)-g(x)}{h}$$

$$=\lim_\limits{h\to0} \dfrac{f(x+h)-f(x)}{h}+\lim_\limits{h\to0} \dfrac{g(x+h)-g(x)}{h}$$

$$=\dfrac{d f(x)}{dx}+\dfrac{d g(x)}{dx}$$

$$=d(f)+d(g)$$