linguistic:translation, structural:summarization
General formula for derivative of implicit function

If R (x, y) = 0, the derivative of the implicit function y (x) is given by

where R and R indicate the partial derivatives of R with respect to x and y.

The above formula comes from using the generalized chain rule to obtain the total derivative — with respect to x — of both sides of R (x, y) = 0:

hence

which, when solved for dy / dx, gives the expression above.