Rules of Differentiation Summary

Calculus 1 · Axiom Academy

SUMMARY Rules of Differentiation Unit 3 recap — the algebra of derivatives: the basic rules, the three combination rules, the derivatives of the special functions, and the two implicit techniques. The basic rules — power, constant, constant-multiple, and sum/difference — let you differentiate any polynomial term by term, without ever returning to the limit definition. The three combination rules handle functions built from simpler ones: the product rule for , the quotient rule for f/g , and the chain rule for a composition f(g(x)) . The chain rule is the engine : the general power rule and every "derivative of a function of x " is just the chain rule applied — always multiply by the derivative of the inside. Memorize the special-function derivatives : , , ; e^x and a^x ; and . These are the atoms every harder derivative is assembled from. When y can't be isolated, use implicit differentiation ; when the function is a tangle of products, quotients, and powers, take a logarithm first and use logarithmic differentiation . The power rule is the workhorse: bring the exponent down and drop it by one. Paired with the constant, constant-multiple, and sum/difference rules, it differentiates any polynomial in one pass. Constant: — a flat line has zero slope. Sum / difference: — differentiate term by term.

This is the written version of the interactive lesson above. See the full Calculus 1 course.