Elasticity (mathematics)
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In mathematics, elasticity of a differentiable function <math>f(x)</math> at point <math>x</math> is defined as
- <math>E(x) = \frac{x f'(x)}{f(x)} = \frac{d \log f(x)}{d \log x}</math>
It is the ratio of the incremental percentage change of the function with respect to an incremental percentage change of the argument. This definition of elasticity is also called point elasticity. If the function is not differentiable the notion of arc elasticity may apply.
If the elasticity is constant <math>E(x) = \alpha</math>, then the function has a form <math>f(x) = C x ^ \alpha</math> for a constant <math>C</math>, which is a solution to the first order differential equation.
The term elasticity has been widely used in economics; see elasticity (economics) for details.
It can be useful to reformulate the above definition so that differentiation is expressed in terms of elasticity
- <math>D f(x) = \frac{E f(x) \cdot f(x)}{x}</math>
Rules for finding the elasticity of products and quotients seem particulary simple, especially as the rule for differentiating quotients is not as simple.
- If <math>h(x) = c</math> then <math>E h(x) = 0\,\!</math>
- If <math>h(x) = x^n</math> then <math>E h(x) = n\,\!</math>
- If <math>h(x) = f(x) \cdot g(x)</math> then <math>E h(x) = E f(x) + E g(x)\,\!</math>
- If <math>h(x) = \frac{f(x)}{g(x)}</math> then <math>E h(x) = E f(x) - E g(x)\,\!</math>
- If <math>h(x) = (f \circ g)(x) = f(g(x))</math> then <math>E h(x) = E f(g(x)) \cdot E g(x)</math>
