Derivatives
Lessons in this chapter
About Derivatives
Derivatives introduces the second foundational idea of calculus: measuring a function’s instantaneous rate of change.
Basics of Derivatives and The First Principle Definition of the Derivative introduce the derivative both informally and through its precise limit-based definition.
Differentiability of a Function at a Point and Continuity and Differentiability cover when a derivative actually exists at a point, and how differentiability and continuity relate (a function must be continuous to be differentiable, but not the reverse).
Laws of Derivatives, Trigonometric Derivatives, and Derivatives of Natural Exponential and Logarithmic Functions close the chapter with the concrete rules for differentiating every basic function type covered so far.
Seven lessons, each a worksheet worked in the page, run from Basics of Derivatives and The First Principle Definition of the Derivative through Laws of Derivatives and Trigonometric Derivatives. The practice problems are scored as you answer them, which is how the derivative formulas end up remembered rather than looked up every time. Free to try, with no login needed for your first 3.
Frequently asked questions
Basics of Derivatives and The First Principle Definition of the Derivative introduce the derivative as a rate of change and then as a limit. Differentiability of a Function at a Point and Continuity and Differentiability cover when a derivative actually exists, and why a function must be continuous to be differentiable but not the other way round. Laws of Derivatives gives the power, product, quotient, and chain rules, and the trigonometric and natural exponential and logarithmic lessons close the chapter with the standard formulas. This is Unit 2 of the AP Calculus AB course framework, Differentiation: Definition and Fundamental Properties, which is past where the Common Core stops.
A derivative measures a function’s instantaneous rate of change at a single point, rather than its average rate of change over an interval.
The first principle definition builds the derivative directly from a limit: the limit, as the interval shrinks to zero, of the average rate of change over that interval.
A function is differentiable at a point when its derivative actually exists there. This requires the graph to have no sharp corner, break, or vertical tangent at that point.
Differentiability requires continuity (a function must have no breaks to be differentiable), but continuity alone doesn’t guarantee differentiability: a continuous graph can still have a sharp corner.
The power rule, product rule, quotient rule, and constant multiple rule give shortcuts for differentiating combinations of functions without going back to the limit definition every time.