Logarithmic Differentiation

Calculus 1 · Axiom Academy

EXAMPLE Logarithmic Differentiation Differentiate a variable raised to a variable power by taking the natural log of both sides first. Find for y = x^ x . The variable appears in both the base and the exponent, so the ordinary power rule and exponential rule don't apply directly — this is exactly where logarithmic differentiation shines. Nice work! You differentiated y = x^ x — a variable raised to a variable power — using logarithmic differentiation. When to use it: The variable sits in both the base and the exponent (or you face a messy product/quotient), so no single power or exponential rule fits. The trick: Take of both sides, then the power law turns an exponent into a product you can differentiate. Implicit differentiation: The left side becomes by the chain rule; the right side uses the product rule. The same take-the-log-then-differentiate playbook handles any "tower" f(x)^ g(x) and tames complicated products and quotients into simple sums.

This is the written version of the interactive lesson above. See the full Calculus 1 course.