Doubling dx and dy

Question

Suppose we have a function $f(x,y)$. Thus the total differential is $df=f_x dx + f_y dy$. Does doubling $dx$ and $dy$ double $df$. I want to multiply everything by $2$ but since we are dealing with infinitesimals I am hesitant to do so.

Answer 1

One way to use the total differential is to create a Taylor series around a point, so $f(x,y)\approx f(x_0,y_0)+f_x(x_0,y_0)(x-x_0)+f_y(x_0,y_0)(y-y_0)$ You can certainly double $x-x_0$ and $y-y_0$ to get the value of $f$ at a different point. Is this what you were talking about?

Answer 2

Think of the total differential as a linear operator: The action of $df(p)$ on a vector of changes $(dx,dy)$ is given by $f_x(p)dx+f_y(p)dy$ where I added a point $p$ to stress that the partial derivatives, hence the differential, depend on some point $p$ where they are evaluated. By linearity, doubling the vector of changes $(dx,dy)$ and then applying the differential will double the result: $$df(p)(2dx,2dy)=f_x(p)(2dx)+f_y(p)(2dy)=2(f_x(p)dx+f_y(p)dy)$$