Mr. Integral

THE BIG IDEA

The Derivative

How can we measure change at a single instant?

Shrink an average-rate interval until its secant becomes a tangent. Once that limit is clear, derivative rules become efficient tools for interpreting graphs and making decisions from models.

Begin the first lesson →See the lesson route ↓

12 lessons · No deadlines or locked lessons

SEE THE CONNECTIONS

How the ideas fit together.

A map of the main ideas. Follow a node to its lesson.

  1. Start with01Instantaneous rate
  2. evaluating the limit efficiently gives02Derivative rules
  3. the sign of the rate explains03Graph behavior
  4. comparing change supports04Optimization

YOUR LEARNING ROUTE

One idea at a time.

Read the explanation, use the visualization, then try the check before opening its reasoning.

Keep the unit cheat sheet handy →
Definition and differentiation methods6 lessons
  1. 01The Derivative from First PrinciplesBuild a derivative from average rates and interpret its units and tangent line.
  2. 02Power Rules and LinearityDifferentiate sums of powers while respecting domains and the meaning of each rule.
  3. 03Product and Quotient RulesChoose a rule from the structure of an expression and simplify without losing domain restrictions.
  4. 04The Chain RuleDifferentiate nested functions by tracking the rate contributed by each layer.
  5. 05Exponential, Logarithmic and Trigonometric DerivativesUse elementary derivative formulas with their domains, angle convention and chain factors.
  6. 06Implicit and Logarithmic DifferentiationDifferentiate relations and variable powers without incorrectly treating dependent variables as constants.
Theorems and graph behavior2 lessons
  1. 07Rolle’s Theorem and the Mean Value TheoremCheck the hypotheses connecting average and instantaneous rates, and use the result to justify monotonicity.
  2. 08Reading Graphs with DerivativesUse sign charts, endpoint checks and concavity to distinguish local and absolute extrema.
Models and approximation4 lessons
  1. 09Optimization from a ModelTranslate constraints into a one-variable objective and justify a global optimum.
  2. 10Related Rates and UnitsDifferentiate a geometric constraint before inserting the measurements of one instant.
  3. 11Linear Approximation and Newton’s MethodUse a tangent line for local estimates and root iterations, while recognizing when either can fail.
  4. 12Indeterminate Forms and L’Hôpital’s RuleApply the derivative-ratio theorem only to valid indeterminate forms and interpret transformed limits.