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I'm currently taking the MIT OCW in multivariate calculus and in the supplementary notes I've seen a notation of partial derivatives that I haven't seen in other textbooks like Thomas' Calculus and I'm confused about the actual meaning of the notation.

Link: https://ocw.mit.edu/courses/mathematics/18-02-multivariable-calculus-fall-2007/readings/tangent_approx.pdf

The notation is $(\frac{\partial f}{\partial x})_0$ to indicate a partial derivative of $f$ respective to $x$ at the point $(x_0 , y_0)$. At first, I thought it meant that the $y$ value must be fixed at $0$, but it didn't seem to be that way. My next theory was that it was supposed to show the subscript of y but in later example it used the same noation for functions of 3 or more variables. So I'm not sure what that $0$ is supposed to represent.

James Lee
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  • Does that OCW allow students to ask questions of the instructor? – GEdgar Jan 03 '21 at 12:18
  • I don't think so.. It is a lecture from 2007 though and the notes are from even before – James Lee Jan 03 '21 at 12:20
  • I'm pretty sure that it follows from the PDF that what is meant is the derivative with respect to $x$ at the point $(x_0,y_0)$. This is pretty wonky notation though... – Jannik Pitt Jan 03 '21 at 12:27
  • So I guess this kind of notation is not used in any other documents? – James Lee Jan 03 '21 at 12:37
  • @JohnWon I have personally never seen that. I think the author wants to emphasise that the "$x$" as the direction in which is differentiated and the point "$x_0$" are different. Often you see something like $\frac{\partial f}{\partial x}(x,y)$ which can be a bit confusing at the beginning because the two "$x$" have different meanings. Both notations just mean the function which assigns every $(x_0,y_0)$ (these can be named arbitrary) the partial derivative in $x$-direction evaluated at $(x_0, y_0)$. – Jannik Pitt Jan 03 '21 at 13:17

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