AP Calculus AB and BC glossary

Power rule

The power rule says that to differentiate a power of the variable, multiply by the exponent and then reduce the exponent by one. It holds for every real exponent, including negative and fractional ones.

\frac{d}{dx}\left[x^n\right] = n x^{n-1}

Its reach is wider than it first appears, because roots and reciprocals are powers in disguise. Writing $\sqrt{x}$ as $x^{1/2}$ and $\frac{1}{x^3}$ as $x^{-3}$ turns both into one-step problems.

The mistake

Applying it to $a^x$ , where the variable is the exponent rather than the base. That is an exponential function and needs a different rule entirely.

Appears in: Unit 2: Defining the Derivative