$\frac{d}{dx}\psi(x,y)=\frac{∂\psi}{∂x}+\frac{∂\psi}{∂y}\frac{dy}{dx}$ .
My interpretation is that the left-hand side is the partial derivative of x.
The right hand is the partial derivative of x, plus the partial derivative of y times the derivative of y with respect to x.
I am absolutely confused. Which is the partial derivative, and what does y have to do with x, aren't they BOTH independent variables??
Doesn't y being expressed as a function of x defeat the entire purpose of a multivariable function??
In my opinion this is much more clear if we don't suppress inputs to functions. Here it is in different notation:
Suppose $f(x) =\Psi(x,y(x))$ for all real numbers $x$. Then $f'(x) = D_1\Psi(x,y(x)) + D_2 \Psi(x,y(x)) y'(x)$ for all real $x$.
Here $D_1 \Psi$ is the partial derivative of $\Psi$ with respect to its first input, and $D_2 \Psi$ is the partial derivative of $\Psi$ with respect to its second input. I am assuming $\Psi$ and $y$ are differentiable functions.