Chain rule for ordinary derivative of function of two variables?

Question

How does this type of chain rule work? I don't see how it makes sense. f is a function of x and y.

df/dx = df/dx + df/dy * dy/dx

Normal "d" is ordinary derivative, italicized "d" is partial derivative.

Answer 1

It is called total derivative and it is just a special case for the chain rule with

$$f=f(t,x(t),y(t),...)$$

More precisely, total derivative is a special case of composition $f\circ g:\mathbb{R}\to\mathbb{R} $ with $f:\mathbb{R^n}\to\mathbb{R}$ and $g:\mathbb{R}\to\mathbb{R^n}$.

In this particular case we have

$$f(x,y(x))\implies \frac{df}{dx}=\frac{\partial f}{\partial x}\frac{\partial x}{\partial x}+\frac{\partial f}{\partial y}\frac{\partial y}{\partial x}=\frac{\partial f}{\partial x}+\frac{\partial f}{\partial y}\frac{dy}{d x}$$

Answer 2

if so then we get $$\frac{\partial F(x,y(x))}{\partial x}+\frac{\partial F(x,y(x))}{\partial y}\cdot\frac{dy(x)}{dx}$$ and this is the chain rule since $$y=y(x)$$