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Maybe it's a stupid question, but if someone could look at this exercise in order to clarify my thoughts. It's from the basics of multivariable calculus: Let $f(x, y, z) = x-2xyz$ where $z = z(x,y)$. The task is to find partial derivatives of $\frac{dz}{dx}$ and $\frac{\partial z}{\partial x}$. The question is: Is it a mistake in this exercise or $\frac{dz}{dx}$ differs from $\frac{\partial z}{\partial x}$? If it does, then how do I compute the result of $\frac{dz}{dx}$? I do know how to do the former one but latter - no clue. Thanks in advance

poyea
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JacekDuszenko
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  • What is $dz/dx$? Even more important: why the Leibniz notation is still used? Only expect an answer to the first question. – Martín-Blas Pérez Pinilla May 06 '18 at 18:46
  • $\partial z/\partial x$ and $\partial z/\partial y$ makes more sense. – Martín-Blas Pérez Pinilla May 06 '18 at 18:49
  • $dz/dx$ is a function that describes rate of change but it makes sense only when z a is one variable function. If it was multi-variable we would have normally used $\frac{\partial z}{\partial x}$, because that is the proper notation for partial derivative, when we want to compute x derivative from z. Therefore, notation $dz/dx$ does not really make sense, am I right? – JacekDuszenko May 06 '18 at 19:54
  • In this context I understand that only $\partial z/\partial x$ makes sense. But maybe via some usual (and confusing) abuse of notation $dz/dx$ also makes sense here. See https://en.wikipedia.org/wiki/Total_derivative. – Martín-Blas Pérez Pinilla May 06 '18 at 20:18
  • Okay, now slightly different question - let's say that I want to compute the $\partial z/\partial x$. Is it a proper notation if I write the chain rule like this? $\partial f/\partial x = \partial f/\partial z * \partial z/\partial x + \partial f/\partial x $ ? – JacekDuszenko May 06 '18 at 20:35
  • and by last $\partial f/\partial x$ I mean that it means it's just a partial derivative of f from x while not considering z=z(x,y). Is this kind of notation correct? In other words the question is: How do I note partial derivative by x when I want to derive only from x and not consider z=z(x,y) in my derivative. – JacekDuszenko May 06 '18 at 20:37
  • What is unclear to me that on my second previous comment I have two identical signs on left and right side of my equation, that is \partial f/\partial x . And I want to say that on the left I take general partial derivative from x. But I've got the same symbols on right and left. How do I tell one from another? – JacekDuszenko May 06 '18 at 20:40
  • About your new question and confusing abuses of notations, see https://math.stackexchange.com/questions/2735523/the-chain-rule-finding-differentials/2735697. The answer to "How do I tell one from another?" is do not us the same name for different things. – Martín-Blas Pérez Pinilla May 06 '18 at 21:13
  • Thank you for your time sir :) – JacekDuszenko May 06 '18 at 23:13

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