calculus - Derivative over variable vs. partial derivative over variable

I was watching one of the Khan Academy videos on differential equations ("Exact Equations Intuition 1 (proofy)") and there's something that confused me.

In the video, they use both the derivative of a function $\psi(x, y(x))$ with respect to $x$,

$$\frac{d}{dx}\psi(x, y(x)),$$

as well as the partial derivative of the same function with respect to the same variable,

$$\frac{\partial \psi}{\partial x}$$

(I think) I understand what a partial derivative of a function is (you consider its other arguments constants and you essentially turn it into a derivative of a single variable function), but I don't understand what a non-partial derivative with respect to one variable means. How is it different from a partial derivative?

Answer

For the partial derivative $\frac{\partial \psi}{\partial x}$, you consider $\psi$ as a function of two independent variables $x$ and $y$, and see how that changes when you vary $x$ while holding $y$ constant. For $\frac{d\psi}{dx}$, you are letting $y$ be a function of $x$ (that's what the $y(x)$ means), so when $x$ changes $y$ also changes. The bivariate version of the chain rule says

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