OCME · Bilingual mathematics & derivations

Concept derivation guides

299 concept teaching guides, separate from the 36,000 individual question derivations.

Concept guide

The derivative: a quadratic's derivative from the difference-quotient limit, and no derivative at a corner

For f(t)=3t²−2t+5: first simplify the difference quotient for h≠0, (f(1+h)−f(1))/h=4+3h, then prove f'(1)=4 by ε-δ; derive the general formulas: the quotient is 2a2x+a1+a2h and the derivative 2a2x+a1; explain why the finite-step quotient 4.3 at h=1/10 is not the derivative, and the affine boundary a2=0; finally prove that g(t)=|t−2| has no derivative at t=2 (not 0), while its derivative is 1 for t>2.

Real derivatives at a point and finite difference quotients

derivation-mko-derivative-01

Concept guide

Derivative rules: checking the product and chain rules against differentiating the expansion

With the ascending coefficient lists f=[2,0,1] and g=[1,3] (that is f(x)=2+x² and g(x)=1+3x): first prove from the definition of the derivative the differentiation formula for polynomials of degree at most three; then find (fg)' and (f∘g)' along two independent routes, differentiating the expansion and applying the product and chain rules, and confirm they agree at every x; also check the sum and scalar rules, expose the errors f'g', a missing inner derivative, an outer derivative at the wrong point and reversed coefficient reading, and evaluate at x=1.

Differentiating polynomial sums, products, and compositions

derivation-mko-derivative-rules-01