Logarithmic differentiation allows us to differentiate additional functions for which other rules may not apply. We start by proving the power rule for arbitrary powers. Using the known differentiation rules and the definition of the derivative, we were only able to prove the power rule in the case of integer powers and the special case of rational powers that were multiples of \(\frac{1}{2}\text{.}\) Logarithmic differentiation gives us a tool that will prove it generally.

Start with the equation \(y = x^p\text{.}\) Create an equivalent equation by taking the logarithm of the absolute value of both sides,

\begin{equation*}
\ln(|y|) = \ln(|x^p|) = \ln(|x|^p).
\end{equation*}

Using the properties of logarithms, this can be rewritten

\begin{equation*}
\ln(|y|) = p \ln(|x|).
\end{equation*}

We now use implicit differentiation and differentiate both sides of the equation:

\begin{gather*}
\frac{d}{dx}[\ln(|y|)] = \frac{d}{dx}[ p \ln(|x|)]\\
\frac{1}{y} \cdot \frac{dy}{dx} = p \cdot \frac{1}{x}.
\end{gather*}

Solving for the derivative, we have

\begin{equation*}
\frac{dy}{dx} = p \cdot \frac{y}{x}.
\end{equation*}

Substituting the original equation \(y=x^p\text{,}\) we obtain the power rule

\begin{equation*}
\frac{dy}{dx} = p x^{p-1}.
\end{equation*}

We can also use logarithmic differentiation to find derivatives of functions represented by powers that are neither power functions nor exponential functions. Recall that a power function must have a constant exponent and an exponential function must have a constant base. If the base and exponent both involve variables, then we are dealing with a function for which we have no differentiation rule.