Subscript notation in partial derivatives [closed]

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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?

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1 Answer

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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!

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