If you assume that y is a function of x (and some other variables), then \partial (y^2)/\partial x= 2y \partial y/\partial x. If y is NOT a function of x, then dy/dx= 0 and so d(y^2)/dx= 0. If you assume that y is a function of the single variable x, then d(y^2)/dx= 2y dy/dx by the chain rule. This has nothing to do with the distinction between "ordinary" and "partial" derivatives. If they did the latter, they would be correct. Contributed by: Joshua Sabloff and Stephen Wang (Haverford College) (March 2011) Open content licensed under CC BY-NC-SA. As either or varies, you can see how a given partial derivative changes both numerically and graphically. Note: They actually use the regular derivative symbols, not the partial derivative symbols. This Demonstration helps students understand what second-order partial derivatives measure. Through my learning of calculus, I have come under the impression that there is an important difference between the derivative of a variable with respect to another, and the partial derivative of a variable with respect to another.
If that's possible, I take this part back. However, you seem to imply that partial derivatives can treat variables dependently. And that as a result, and because of the fact that y's independence of x in regular derivatives is rightly in question, the regular derivative calculation should at the very least give the safe, always correct answer.
I would have also made the point that we actually have the notation and convention for deriving a variable independently with relation to another, and that that is the partial derivative. My proposition is that dy^2/dx simply return 2y*dy/dx which is always correct! Whether one wishes to make an assumption (and it is an assumption) that y is independent of x, should be one's own decision to make, not the calculator's. The order of operations convention isn't a very good comparison to make, since the answer the calculator returns there is either entirely correct or entirely wrong in regards to the user's intention (and that intention misplaced if he disagrees with order of operations, a very strong convention) The user that does not adust for it is incorrect. The dy/dx = 0 assumption is not incorrect.
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