Applications of Derivatives
Lessons in this chapter
About Applications of Derivatives
Applications of Derivatives puts every derivative rule to work solving real, practical problems.
Interpreting Derivatives in a Context reads a derivative’s real-world meaning (like velocity from a position function), and Related Rates covers finding how one changing quantity’s rate relates to another’s.
Tangents and Normals and Linear Approximations use the derivative to find a tangent line at a point and to estimate a function’s value nearby using that tangent line.
L’Hospital’s Rule closes the chapter with a derivative-based shortcut for evaluating limits that would otherwise come out as indeterminate forms like 0/0.
Each of the five lessons, including Related Rates, Tangents and Normals, and L'Hospital's Rule, opens as a practice worksheet worked in the page. The problems are scored as you answer them, which is the point with related rates, where setting the equation up is most of the work. Free to try, with no login needed for your first 3.
Frequently asked questions
Interpreting Derivatives in a Context reads a derivative as a real rate, with units, rather than as an expression to simplify. Related Rates links two quantities changing together and solves for one rate from the other, Tangents and Normals finds the lines touching and perpendicular to a curve at a point, Linear Approximations uses a tangent line to estimate nearby values, and L'Hospital's Rule evaluates indeterminate limits using derivatives. This is Unit 4 of the AP Calculus AB course framework, Contextual Applications of Differentiation, and it goes past what the Common Core covers.
A derivative’s real-world meaning depends on context, as a rate of change of position, it’s velocity; as a rate of change of velocity, it’s acceleration.
A related rates problem uses the chain rule to connect the rates of change of two or more quantities that are related by an equation, like a ladder sliding down a wall.
A tangent line touches a curve at one point with the same slope as the curve there (given by the derivative), while a normal line is perpendicular to that tangent at the same point.
A linear approximation uses a function’s tangent line at a known point to estimate the function’s value at a nearby point, without evaluating the function directly.
L'Hospital's Rule evaluates a limit that comes out as an indeterminate form (like 0/0) by taking the derivative of the numerator and denominator separately, then trying the limit again.