Logarithmic Differentiation

The power rule $\dfrac{d}{dx}(x^n) = nx^{n-1}$ applies when the exponent is a fixed number, and the exponential rule $\dfrac{d}{dx}(a^x) = a^x \log a$ applies when the base is a fixed positive number other than 1. Neither rule covers a function where both the base and the exponent vary with $x$ — something of the form $[f(x)]^{g(x)}$, such as $x^x$ or $\left(x + \dfrac{1}{x}\right)^{x-1}$.

For functions of this type, the trick is to take the logarithm of both sides first, which turns the troublesome variable exponent into a product via $\log(a^b) = b\log a$, and only then differentiate implicitly. For instance, writing $y = x^x$ as $\log y = x \log x$ and differentiating both sides with respect to $x$ gives $\dfrac{1}{y}\dfrac{dy}{dx} = \log x + 1$, so $\dfrac{dy}{dx} = x^x(\log x + 1)$. …