Step By Step Calculus » 11.5 - Implicit Differentiation using Partial Derivative Technique

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Synopsis
Implicit Differentiation is the method to find \displaystyle \frac{dy}{dx}\displaystyle \frac{dy}{dx} from an equation of the form an implicitly defined function (i.e. f(x,y)=af(x,y)=a) between xx and yy. In doing Implicit Differentiation, one will have to treat yy as a function of xx and then differentiate the equation f(x,y)=af(x,y)=a using the chain rule.
A quicker method to find \displaystyle \frac{dy}{dx}\displaystyle \frac{dy}{dx} involves partial differentiation as follows:
\displaystyle \frac{dy}{dx}=-\frac{\frac{\partial}{\partial x}f(x,y)}{\frac{\partial}{\partial y}f(x,y)}\qquad \displaystyle \frac{dy}{dx}=-\frac{\frac{\partial}{\partial x}f(x,y)}{\frac{\partial}{\partial y}f(x,y)}\qquad provided \displaystyle \frac{\partial }{\partial y}f(x,y)\neq 0. \displaystyle \frac{\partial }{\partial y}f(x,y)\neq 0.