|
Size: 691
Comment: Example
|
Size: 1397
Comment: More detail
|
| Deletions are marked like this. | Additions are marked like this. |
| Line 27: | Line 27: |
| [[Economics/DemandCurve#Point_Elasticity|Point elasticity]] can be differentiated with the chain rule: | [[Economics/DemandCurve#Point_Elasticity|Point elasticity]] is defined as: |
| Line 31: | Line 31: |
| It can be simplified using the chain rule into: |
|
| Line 32: | Line 34: |
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'' 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'' 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 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 with respect to Q is 1/Q. This leaves the equation as:
