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^{2}+ 2y

^{2}and [tex] x = rcos(\theta), y = rsin(\theta) [/tex].

i have [tex] \frac{\partial f}{\partial y} [/tex] (while holding x constant) [tex] = 4y [/tex]. and [tex] \frac{\partial f}{\partial y} [/tex] (while holding r constant) [tex] = 2y [/tex].

i found these partial derivatives by expressing f in terms of only x and y, and then in terms of only r and y. But i am sure there are times where it can be very difficult to solve for one variable or to express some function in terms of specific variables.

Is there a way to relate the 2 partial derivatives with respect to y (one holding x constant and one holding r constant) using the chain rule or something?