Topic 2 of 6
Derivatives
Rules, structure, and the meaning of instantaneous change—from the chain rule to implicit curves.
01Calculus · Direct answerWhat is the derivative of x to the x?The base and exponent both vary, so neither the ordinary power rule nor the simple exponential rule works alone.02Calculus · Method guideThe Chain Rule as a structure-reading skillDifferentiate the outside function, keep the inside intact, then multiply by the inside rate.03Calculus · Method guideImplicit differentiation without losing dy/dxDifferentiate an equation whose y-values are not isolated, treating y as a function of x every time it appears.04Calculus · Method guideWhen logarithmic differentiation is the clean moveUse logarithms to untangle variable exponents and products or quotients with many factors.05Calculus · Decision guideProduct Rule or Quotient Rule?Choose the derivative rule from the top-level operation—and simplify first when algebra can remove the choice entirely.