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how do you read partial derivatives? I have seen people reading from left to right, and have seen people reading from right to left.for example what does this equation mean $\frac{\partial f}{\partial x \partial y}$:

1)at first you differentiate with respect to x and then y, or

2)at first you differentiate with respect to y and then with respect to x?

earlier I knew the first one, now I have read the authors used 2).

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1 Answer 1

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The standard notation is the $2)$: $$ \frac{\partial^2 f}{\partial x \partial y} := \frac{\partial}{\partial x} \left(\frac{\partial f}{\partial y}\right) $$

EDIT

Other notations:

  • $f_{yx}=\frac{\partial^2 f}{\partial x \partial y}$
  • $\partial_x \partial_y f=\frac{\partial^2 f}{\partial x \partial y} $
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    $\begingroup$ And $\frac{\partial}{\partial x}(\frac{\partial f}{\partial y})=f_{yx}$, right? $\endgroup$
    – bof
    Jun 13, 2020 at 11:03
  • $\begingroup$ @bof Yes: $$f_{yx}= (f_y)_x=\frac{\partial}{\partial x} f_y =\frac{\partial}{\partial x} \frac{\partial f}{\partial y} = \frac{\partial^2 f}{\partial xy} $$ $\endgroup$
    – Sewer Keeper
    Jun 13, 2020 at 11:06

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