AP Calculus AB and BC glossary

Logarithmic differentiation

Logarithmic differentiation is a technique where you take the natural logarithm of both sides of an equation before differentiating implicitly. It converts products into sums and exponents into coefficients, which is what makes a variable base raised to a variable power differentiable.

It is the only reasonable route for something like $y = x^x$ , where neither the power rule nor the exponential rule applies because the variable appears in both the base and the exponent.

The steps are fixed: take $\ln$ of both sides, use log properties to break the expression apart, differentiate implicitly, then multiply through by $y$ and substitute the original expression back in.

Do not forget the last step

Implicit differentiation leaves you with $\frac{dy}{dx}$ in terms of $y$ . The answer is not finished until you replace $y$ with the original function of $x$ .

Appears in: Unit 3: Chain Rule, Implicit, and Inverses