Rules of differentiation
Product rule: first times derivative of second plus second times derivative of first; the chain rule multiplies by the inside derivative.
A worked example
For f(x) = (x³ + 2)⁴ the chain rule gives f′(x) = 4(x³ + 2)³ x 3x² = 12x²(x³ + 2)³, so f′(2) = 12 x 4 x 1 000 = 48 000.
The mistake people actually make
The quotient rule subtracts in a fixed order: bottom times derivative of top, minus top times derivative of bottom, all over the bottom squared. The inside derivative in a chain rule is the piece most often dropped.