Unit 3 of 10

Derivatives

  • 13 lessons
  • Parts 39 to 64

The derivative measures how fast a quantity changes at a single instant. Geometrically it is the slope of the tangent line, and physically it is velocity, growth rate or marginal cost, depending on what the function describes.

This unit defines the derivative as a limit, then develops the rules that let you differentiate almost any formula quickly, including the product, quotient and chain rules and the derivatives of trigonometric, exponential, logarithmic, inverse and hyperbolic functions.

Every lesson is a pair of videos. The theory part explains the idea and the practice part works through problems. Each lesson page has the definitions and an example for every key term.

Definition of the Derivative

  1. Derivatives and Rates of Change

    The derivative of a function at a point as the limit of difference quotients, read as the slope of the tangent line, an instantaneous rate of change and a velocity, with its units.

    Theory part 39Practice part 40

  2. The Derivative as a Function

    How the derivative becomes a new function, the notations used for it, what differentiability on an interval means, why it implies continuity, and how corners, cusps and vertical tangents break it.

    Theory part 41Practice part 42

Differentiation Rules

  1. Elementary Derivative Rules

    The constant, power, constant multiple, sum and difference rules, where the power rule is valid on the real line, and how they give the derivative of any polynomial and of the absolute value away from zero.

    Theory part 43Practice part 44

  2. The Product Rule

    Why the derivative of a product is not the product of the derivatives, what the correct rule is, how a growing rectangle explains it, how to prove it, and how it extends to three or more factors.

    Theory part 45Practice part 46

  3. The Quotient Rule

    How to differentiate a reciprocal and a quotient of differentiable functions, why the denominator must be nonzero, and how the rule follows from the product rule and applies to every rational function.

    Theory part 47Practice part 48

  4. The Chain Rule

    How to differentiate a function of a function by multiplying the rates of the outer and inner parts, how this handles nested compositions and related rates, and which differentiability conditions it needs.

    Theory part 49Practice part 50

Derivatives of Elementary Functions

  1. Derivatives of Trigonometric Functions

    Why the derivative of sine is cosine when angles are in radians, and how the derivatives of all six trigonometric functions follow from that one fact.

    Theory part 51Practice part 52

  2. Derivatives of Exponential and Logarithmic Functions

    Why the natural exponential function is its own derivative, how that gives the derivatives of other exponentials and of logarithms, and on which domains each formula holds.

    Theory part 53Practice part 54

Differentiation Rules

  1. Higher-Order Derivatives

    How to differentiate a derivative again, what second and third derivatives mean for motion, how to differentiate a product many times, and what smoothness classes measure.

    Theory part 55Practice part 56

  2. Implicit Differentiation

    How to find the slope of a curve given by an equation in x and y without solving for y, when that slope makes sense, and how to get second derivatives the same way.

    Theory part 57Practice part 58

Derivatives of Elementary Functions

  1. Derivatives of Inverse Functions

    How the slope of an inverse function comes from the slope of the original, why that slope must be nonzero, and how this gives the derivatives of arcsin, arccos and arctan.

    Theory part 59Practice part 60

  2. Logarithmic Differentiation

    How taking a logarithm before differentiating turns long products and quotients into sums, handles functions like x to the power x, and which sign and domain conditions it needs.

    Theory part 61Practice part 62

  3. Derivatives of Hyperbolic Functions

    What sinh, cosh and tanh are, why their derivatives look like trigonometric ones with different signs, and how to differentiate their inverses on the right real domains.

    Theory part 63Practice part 64