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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. 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}} |
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:
Usage
Point elasticity is defined as:
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. The derivative of e to an exponent is itself, so the derivative of P = ev 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:
