Elasticity of a function

In mathematics, elasticity of a positive differentiable function "f" at point "x" is defined as:$Ef(x)\; =\; frac\{x\}\{f(x)\}f\text{'}(x)\; =\; frac\{d\; log\; f(x)\}\{d\; log\; x\}$It is the ratio of the incremental change of the logarithm of a function with respect to an incremental change of the logarithm of the argument. This definition of elasticity is also called point elasticity, and is the limit of arc elasticity between two points.

The term elasticity is widely used in economics; see elasticity (economics) for details.

Rules

Let "a" be a constant. Then:$E\; a\; =\; 0$, :$E\; a\; f(x)\; =\; a\; E\; f(x)$, and:$E\; x^a\; =\; a.$Any function f has constant point elasticity if and only if it is a monomial of the form $ax^b$.

Although elasticity is not a linear operator like the derivative, rules for finding the elasticity of products and quotients are simpler than those for derivatives. Let "f, g" be differentiable. Then:$E\; f(x)\; cdot\; g(x)\; =\; E\; f(x)\; +\; E\; g(x).$:$E\; frac\{f(x)\}\{g(x)\}\; =\; E\; f(x)\; -\; E\; g(x).$:$E\; (f\; circ\; g)(x)\; =\; E\; f(g(x))\; cdot\; E\; g(x).$

The differentiation can be expressed in terms of elasticity as:$D\; f(x)\; =\; frac\{E\; f(x)\; cdot\; f(x)\}\{x\}$

References

* Yves Nievergelt, The Concept of Elasticity in Economics, "SIAM Review", Vol. 25, No. 2 (Apr., 1983), pp. 261-265