You have formulas that allow you to differentiate \(x^2\text{,}\) \(2^x\text{,}\) and \(2^2\text{.}\) You don't, however, have a formula to differentiate \(x^x\text{.}\) In this section you are going to use a process called logarithmic differentiation to determine the derivative formula for the function \(y=x^x\text{.}\) Example 7.3.1 shows this process for a different function.

With logarithmic differentiation, when \(y\) is written as a function of \(x\text{,}\) especially one involving only products, quotients, and powers, begin by taking the logarithm of both sides of the equation relating \(y\) to \(x\text{.}\) This technique will usually only be helpful if the logarithm rules allow you to break up the right side some. Proceed to differentiate each side with respect to \(x\text{.}\) On the left side, respect the rules for implicit differentiation. Once this is done, algebraically solve for \(\lz{y}{x}\text{.}\) If there are any instances of \(y\) left, replace them with the original expression in \(x\text{.}\)

In the olden days (before symbolic calculators) we would use the process of logarithmic differentiation to find derivative formulas for complicated functions. The reason this process is “simpler” than straight forward differentiation is that we can obviate the need for the product and quotient rules if we completely expand the logarithmic expression before taking the derivative.