Intuition behind the formula of total differential

Question

Formula of total differential: $dy(x_1, ..., x_n) = $${\partial f\over\partial x_1}dx_1+\cdots+{\partial f\over\partial x_n}dx_n$

I can not understand why there are specifically addition of ${\partial f\over\partial x_m}dx_m$ monomials, not say, subtraction or multiplication

Answer 1

How much does $f$ change as its input changes from $x$ to $x + \Delta x$?

You can move from $x$ to $x + \Delta x$ by the following sequence of steps. (I'll assume $n = 3$ for simplicity.)

1. Move from $(x_1,x_2,x_3)$ to $(x_1 + \Delta x_1,x_2,x_3)$.
2. Move from $(x_1 + \Delta x_1,x_2,x_3)$ to $(x_1 + \Delta x_1,x_2 + \Delta x_2,x_3)$.
3. Move from $(x_1 + \Delta x_1,x_2 + \Delta x_2,x_3)$ to $(x_1 + \Delta x_1,x_2 + \Delta x_2,x_3 + \Delta x_3)$.

Now let's estimate how much $f$ changes at each step, assuming that $\Delta x$ is very small.

1. The change is approximately $D_1 f(x_1,x_2,x_3) \Delta x_1$.
2. The change is approximately $D_2 f(x_1 + \Delta x_1,x_2,x_3) \Delta x_2$.
3. The change is approximately $D_3 f(x_1 + \Delta x_1,x_2 + \Delta x_2,x_3) \Delta x_3$.

If we now make the approximation that $$ D_2 f(x_1 + \Delta x_1,x_2,x_3) \approx D_2 f(x_1,x_2,x_3) $$ and $$ D_3 f(x_1 + \Delta x_1,x_2 + \Delta x_2,x_3) \approx D_3 f(x_1,x_2,x_3) $$ then we obtain $$ \Delta f \approx D_1 f(x_1,x_2,x_3) \Delta x_1 + D_2 f(x_1,x_2,x_3) \Delta x_2 + D_3 f(x_1,x_2,x_3) \Delta x_3. $$

Edit 1: Here is some more detail about how I estimated the change in $f$ at step 1. The partial derivative $D_1 f(x)$ is defined by $$ D_1 f(x) = \lim_{\Delta x_1 \to 0} \frac{f(x_1 + \Delta x_1,x_2,x_3) - f(x_1,x_2,x_3)}{\Delta x_1}. $$ It follows that $$ D_1 f(x) \approx \frac{f(x_1 + \Delta x_1,x_2,x_3) - f(x_1,x_2,x_3)}{\Delta x_1} $$ when $\Delta x_1$ is close to $0$. And from this it follows that $$ f(x_1 + \Delta x_1,x_2,x_3) \approx f(x_1,x_2,x_3) + D_1 f(x) \Delta x_1 $$ when $x_1$ is close to $0$. (This statement is one of the key pieces of intuition behind calculus, and is worth internalizing.) In other words, when the input to $f$ changes from $(x_1,x_2,x_3)$ to $(x_1 + \Delta x_1,x_2,x_3)$, the value of $f$ changes by approximately $D_1 f(x) \Delta x_1$.