Applications of Derivatives Summary

Calculus 1 · Axiom Academy

SUMMARY Applications of Derivatives One toolkit, seven uses: the derivative measures change, so it solves rates, extremes, shapes, and fast approximations. A derivative is a rate of change , so every application here is really one idea worn seven ways. Critical points ( f'(x)=0 or undefined) are where extremes can live; the sign of f'' tells max from min. The second derivative governs concavity and curve shape; f'' sign changes mark inflection points. The tangent line is the best linear stand-in for a curve nearby — linearization and Newton's method both run on it. Big-theorem guarantees — the Mean Value Theorem and L'Hôpital's Rule — let derivatives certify behavior and crack 0/0 limits. When two quantities are linked by an equation and both change over time, differentiating that equation implicitly with respect to t relates their rates. Solve for the rate you want. When to use: one rate is given, another is asked, and a geometric or physical relation ties the variables. Watch out for: substitute fixed numbers after differentiating, never before — a constant has rate zero. To maximize or minimize, reduce the objective to one variable using the constraint, set the derivative to zero, then confirm with the second-derivative test or by comparing endpoints. When to use: "largest / smallest / cheapest / fastest" with a constraint connecting the variables. Watch out for: on a closed interval the answer can sit at an endpoint, not a critical point — always check both.

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