Topic 2 of 6
Derivatives
Rules, structure, and the meaning of instantaneous change—from the chain rule to implicit curves.
01Direct 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.02Method guideThe Chain Rule as a structure-reading skillDifferentiate the outside function, keep the inside intact, then multiply by the inside rate.03Method 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.04Method guideWhen logarithmic differentiation is the clean moveUse logarithms to untangle variable exponents and products or quotients with many factors.05Decision guideProduct Rule or Quotient Rule?Choose the derivative rule from the top-level operation—and simplify first when algebra can remove the choice entirely.