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implicit differentiation
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(Definition)
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Implicit differentiation is a tool used to analyze functions that cannot be conveniently put into a form
where
. To use implicit differentiation meaningfully, you must be certain that your function is of the form
(it can be written as a level set) and that it satisfies the implicit function theorem ( must be continuous, its first partial derivatives must be continuous, and the derivative with respect to the implicit function must be non-zero). To actually differentiate implicitly, we use the chain rule to differentiate the entire equation.
Example: The first step is to identify the implicit function. For simplicity in the example, we will assume and is an implicit function of . Let
(Since this is a two dimensional equation, all one has to check is that the graph of may be an implicit function of in local neighborhoods.) Then, to differentiate implicitly, we differentiate both sides of the equation with respect to . We will get
Do you see how we used the chain rule in the above equation ? Next, we simply solve for our implicit derivative
. Note that the derivative depends on both the variable and the implicit function . Most of your derivatives will be functions of one or all the variables, including the implicit function itself.
[better example and ?multidimensional? coming]
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"implicit differentiation" is owned by slider142.
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(view preamble)
Cross-references: variable, sides, neighborhoods, graph, equation, entire, chain rule, differentiate, derivative, partial derivatives, continuous, implicit function theorem, level set, functions
There are 6 references to this entry.
This is version 2 of implicit differentiation, born on 2002-02-25, modified 2003-10-30.
Object id is 2660, canonical name is ImplicitDifferentiation.
Accessed 14403 times total.
Classification:
| AMS MSC: | 26B10 (Real functions :: Functions of several variables :: Implicit function theorems, Jacobians, transformations with several variables) |
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Pending Errata and Addenda
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