![Subscript Notation in Partial Derivatives [Closed]](https://docuprint.cloud68.co/uploads/subscript-notation-in-partial-derivatives-closed_1200.jpg)
just want to be clear with partial derivative notation, and as to what respect we are taking the partial deriv of , for example; $$f_x, f_y, f_{xx}, f_{yy}, f_{xy}$$
so with what respect is each partial deriv being taken of? are they all with respect to x?
1 Answer
Let $f \colon \mathbb{R}^2 \to \mathbb{R}$, $f \in C^{k}(\mathbb{R}^2)$. Then $$f_x(x_0,y_0) = \frac{\partial f}{\partial x}(x_0, y_0)$$ $$f_y(x_0,y_0) = \frac{\partial f}{\partial y}(x_0, y_0)$$ $$(f_x)_x(x_0,y_0) = f_{xx}(x_0,y_0) =\frac{\partial \Big(\frac{\partial f}{\partial x}\Big)}{\partial x}(x_0,y_0) = \frac{\partial f_x}{\partial x}(x_0,y_0) = \frac{\partial^2 f}{\left.\partial x\right.^2}(x_0, y_0)$$ $$(f_x)_y(x_0,y_0) = f_{xy}(x_0,y_0) =\frac{\partial \Big(\frac{\partial f}{\partial x}\Big)}{\partial y}(x_0,y_0) = \frac{\partial f_x}{\partial y}(x_0,y_0) = \frac{\partial^2 f}{\partial y \partial x}(x_0, y_0)$$
Is it clear now? I hope so!
