AP Calculus AB and BC glossary
Outer function
The outer function is the one applied last in a composition, wrapping around everything else. The chain rule differentiates it first, evaluated at the inner function rather than at x, and then multiplies by the inner function's derivative.
The order of operations you would use to EVALUATE the expression identifies the layers. For you take the logarithm first and cube last, so cubing is the outer function.
The mistake
Substituting x into the outer derivative. It must be evaluated at the inner function, so the derivative of the cube of ln x contains ln x squared, not x squared.
Appears in: Unit 3: Chain Rule, Implicit, and Inverses