Differences between revisions 2 and 3

Chain Rule

The chain rule is an approach for differentiating a composition of functions in terms of those functions.

Contents

Notation

The common notation for the chain rule is, given a function h(x) = f(g(x)), the derivative h'(x) = f'(g(x)) g'(x).

An equivalent notations is that h = f ∘ g and h' = (f ∘ g)' = (f' ∘ g)g' .

A parallel notation is:

It can be simplified using the chain rule into:

Start with the final formulation and use u- and v-substitution:

Note in particular that the equation for v can be rewritten in terms of P:

Now the final formulation can be rewritten as:

Several of these terms are trivial to derive.

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  Comment: Example
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-[[Economics/DemandCurve#Point_Elasticity|Point elasticity]] can be differentiated with the chain rule:
+[[Economics/DemandCurve#Point_Elasticity|Point elasticity]] is defined as:
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+It can be simplified using the chain rule into:
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+Start with the final formulation and use ''u''- and ''v''-substitution:

{{attachment:substitution1.svg}}

Note in particular that the equation for ''v'' can be rewritten in terms of ''P'':

{{attachment:substitution2.svg}}

Now the final formulation can be rewritten as:

{{attachment:substitution3.svg}}

Several of these terms are trivial to derive.