Logarithmic Differentiation

Logarithmic differentiation takes the (natural) logarithm of both sides before differentiating. It is the right tool in two situations:

1. A variable raised to a variable power, $y = [f(x)]^{g(x)}$ — for example $y = x^x$. Here neither the power rule ($x^n$ needs a constant $n$) nor the exponential rule ($a^x$ needs a constant base $a$) applies, because both base and exponent vary. Taking logs converts the awkward power into a product:

   $y = [f(x)]^{g(x)} \;\Rightarrow\; \ln y = g(x)\,\ln f(x),$

   which is then differentiated using the product and chain rules.
2. A long product or quotient of many factors — logs turn products into sums and quotients into differences, so $\ln y$ becomes a simple sum that differentiates term by term, avoiding a messy product/quotient-rule computation. The key step — differentiating $\ln y$. Since $y$ is a function of $x$, the chain rule gives

   $\frac{d}{dx}\big(\ln y\big) = \frac{1}{y}\,\frac{dy}{dx}.$

   So after differentiating both sides you isolate $\dfrac{dy}{dx}$ by multiplying through by $y$, then substitute the original expression for $y$. Method — logarithmic differentiation:

1. Write $y =$ the expression; take natural logs of both sides.
2. Use log laws to expand: $\ln(ab)=\ln a+\ln b$, $\ln\tfrac{a}{b}=\ln a-\ln b$, $\ln(a^n)=n\ln a$.
3. Differentiate both sides with respect to $x$; the left side gives $\dfrac{1}{y}\dfrac{dy}{dx}$.
4. Multiply both sides by $y$ and replace $y$ by the original expression.