= Chain Rule =

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

<<TableOfContents>>

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== 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:

{{attachment:leibniz.svg}}

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== Usage ==

[[Economics/DemandCurve#Point_Elasticity|Point elasticity]] is defined as:

{{attachment:elasticity1.svg}}

It can be simplified using the chain rule into:

{{attachment:elasticity2.svg}}

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. The derivative of ''e'' to an exponent is itself, so the derivative of ''P = e^v^'' with respect to ''v'' is ''P''. The derivative of a natural logarithm of some variable is 1 over that variable, so the derivative of ''u = ln Q'' with respect to ''Q'' is ''1/Q''. This leaves the equation as:

{{attachment:substitution4.svg}}



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CategoryRicottone