I know the equation for economic elasticity is:
$$\varepsilon = \frac{\%\,\Delta Y}{\%\,\Delta X}\frac{X}{Y} = \frac{\partial Y(X)}{\partial X}\frac{X}{Y} = \frac{\partial \log(Y)}{\partial \log(X)}$$
In fact, this is the generalization for any sensitive-analysis or elasticity for a given function; in this case $Y(X)$.
But, where does come it from? I mean, why is it the equation and no -for example- just the partial derivate of the given function?
Thanks in advance!
The elasticity gives you the percentage change of the dependent variable with respect to the percentage change of the independent variable. Elasticity by definition is dimensionless and I believe this is part of the motivation behind using this over the derivative like you mentioned.