differential equations for xx

In this entry, we will derive differential equationsMathworldPlanetmath satisfied by the function xx. 11In this entry, we restrict x, and hence xx to be strictly positive real numbers, hence it is justified to divide by these quantities. We begin by computing its derivativePlanetmathPlanetmath. To do this, we write xx=exlogx and apply the chain ruleMathworldPlanetmath:


Set y=xx. Then we have y/y=1+logx. Taking another derivative, we have


Applying the quotient ruleMathworldPlanetmath and simplifying, this becomes


It is also possible to derive an equation in which x does not appear. We start by noting that, if z=1/x, then z+z2=0. If, as above, y=xx, we have (d/dx)(y/y)=z. Combining equations,


applying the quotient rule and simplifying,

Title differential equations for xx
Canonical name DifferentialEquationsForXx
Date of creation 2013-03-22 17:24:37
Last modified on 2013-03-22 17:24:37
Owner rspuzio (6075)
Last modified by rspuzio (6075)
Numerical id 6
Author rspuzio (6075)
Entry type DerivationMathworldPlanetmath
Classification msc 26A99