Elasticity of a function
In mathematics, the elasticity or point elasticity of a positive differentiable function f of a positive variable (positive input, positive output) at point a is defined as E f ( a ) = a f ( a ) f ′ ( a ) {\displaystyle Ef(a)={\frac {a}{f(a)}}f'(a)} = lim x → a f ( x ) − f ( a ) x − a a f ( a ) = lim x → a f ( x ) − f ( a ) f ( a ) a x − a = lim x → a f ( x ) f ( a ) − 1 x a − 1 ≈ % Δ f ( a ) % Δ a {\displaystyle =\lim _{x\to a}{\frac {f(x)-f(a)}{x-a}}{\frac {a}{f(a)}}=\lim _{x\to a}{\frac {f(x)-f(a)}{f(a)}}{\frac {a}{x-a}}=\lim _{x\to a}{\frac {{\frac {f(x)}{f(a)}}-1}{{\frac {x}{a}}-1}}\approx {\frac {\%\Delta f(a)}{\%\Delta a}}} or equivalently E f ( x ) = d log f ( x ) d log x . {\displaystyle Ef(x)={\frac {d\log f(x)}{d\log x}}.} It is thus the ratio of the relative (percentage) change in the function's output f ( x ) {\displaystyle f(x)} with respect to the relative change in its input x {\displaystyle x} , for infinitesimal changes from a point ( a , f ( a ) ) {\displaystyle (a,f(a))} .