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
$$ \frac{d}{dx} \psi(x,y(x)) = \frac{\partial \psi}{\partial x}(x,y(x)) +
\frac{dy}{dx}(x) \frac{\partial \psi}{\partial y}(x,y(x))$$
No comments:
Post a Comment