Understanding the Chain Rule: (df/dx) + (df/dy)* (dy/dx)

[Joseph1739]

(df/dx) + (df/dy)* (dy/dx) = df(x,y)/dx

My book mentions the chain rule to obtain the right side of the equation, but I don't see how. The chain rule has no mention of addition. The furthest I got was applying the chain rule to the right operant resulting in:

df/dx + df/dx = 2(df/dx)

 

[Samy_A]

(df/dx) + (df/dy)* (dy/dx) = df(x,y)/dx

My book mentions the chain rule to obtain the right side of the equation, but I don't see how. The chain rule has no mention of addition. The furthest I got was applying the chain rule to the right operant resulting in:

df/dx + df/dx = 2(df/dx)

I'm trying to make sense of this equation.

The only thing I can think of is this:
$f$ is a function of two variables, $x$ and $y$, and then they consider that $y$ is some function of $x$.
What they actually compute is the derivative of $g(x)=f(x,y(x))$.
For that, the chain rule can be used, giving:
$\frac{dg}{dx}=\frac{\partial f}{\partial x}+\frac{\partial f}{\partial y} \frac{dy}{dx}$.

It's a little confusing, as $y$ is both the name of the second variable and considered a function of $x$. And they don't use a new name $g$, but keep $f$.
That's how they get $\frac{df}{dx}=\frac{\partial f}{\partial x}+\frac{\partial f}{\partial y} \frac{dy}{dx}$.

It's somewhat sloppy, as they don't use $\partial$ for what clearly must be partial derivatives.