implicit differentiation

Implicit differentiationMathworldPlanetmath is a tool used to analyze the solution sets of equations of the form f(𝐱,𝐲)=0 that cannot be conveniently put into a form 𝐲=g(𝐱) that explicitly shows the dependence of 𝐲 on the variable 𝐱. To use implicit differentiation meaningfully, you must be certain that 𝐲 is actually a functionMathworldPlanetmath of 𝐱 in a neighborhoodMathworldPlanetmath of the point (𝐚,𝐛) where you plan to apply the derivative. This means you want to ensure that f(𝐱,𝐲) satisfies the implicit function theoremMathworldPlanetmath in that neighborhood (f must be continuously differentiable at (𝐚,𝐛), and its derivative must be non-singular at (𝐚,𝐛)). If this is not the case, the quantity you get from naively differentiating both sides of the equation may not represent the derivative. To actually differentiate implicitly, we use the chain ruleMathworldPlanetmath to differentiate the entire equation.

Example: The first step is to identify the implicit function. Suppose we have simplified an equation to the form x2+y2+xy=0 (Since this is a two dimensional equation, all one has to check is that the graph of y may be an implicit function of x in local neighborhoods.) Then, to differentiate implicitly, we differentiate both sides of the equation with respect to x, using the chain rule whenever we encounter y, as y is treated as a function of x. We will get


Next, we simply solve for our implicit derivative dydx=-2x+y2y+x. Note that the derivative depends on both the variable and the implicit function y. Most of your derivatives will be functions of one or all the variables, including the implicit function itself.

Title implicit differentiation
Canonical name ImplicitDifferentiation
Date of creation 2013-03-22 12:28:22
Last modified on 2013-03-22 12:28:22
Owner slider142 (78)
Last modified by slider142 (78)
Numerical id 8
Author slider142 (78)
Entry type Definition
Classification msc 26B10