Editing Implicit function (section)

===General formula for derivative of implicit function===
If {{math|1=''R''(''x'', ''y'') = 0}}, the derivative of the implicit function {{math|''y''(''x'')}} is given by<ref name="Stewart1998">{{cite book | last = Stewart | first = James | title = Calculus Concepts And Contexts | publisher = Brooks/Cole Publishing Company | year = 1998 | isbn = 0-534-34330-9 | url-access = registration | url = https://archive.org/details/calculusconcepts00stew }}</ref>{{rp|§11.5}}

:<math>\frac{dy}{dx} = -\frac{\,\frac{\partial R}{\partial x}\,}{\frac{\partial R}{\partial y}} = -\frac {R_x}{R_y} \,,</math>

where {{math|''R<sub>x</sub>''}} and {{math|''R<sub>y</sub>''}} indicate the [[partial derivative]]s of {{mvar|R}} with respect to {{mvar|x}} and {{mvar|y}}.

The above formula comes from using the [[Chain rule#Multivariable case|generalized chain rule]] to obtain the [[total derivative]] — with respect to {{mvar|x}} — of both sides of {{math|1=''R''(''x'', ''y'') = 0}}:

:<math>\frac{\partial R}{\partial x} \frac{dx}{dx} + \frac{\partial R}{\partial y} \frac{dy}{dx} = 0 \,,</math>

hence

:<math>\frac{\partial R}{\partial x} + \frac{\partial R}{\partial y} \frac{dy}{dx} =0 \,,</math>

which, when solved for {{math|{{sfrac|''dy''|''dx''}}}}, gives the expression above.