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Differential Calculus

A brief introduction to differential calculus

Watch an introduction video9 minutes 7 seconds
How would you like to follow in the footsteps of Euclid and Archimedes? Would you like to be able to determine precisely how fast Usain Bolt is accelerating exactly 2 seconds after the starting gun? Differential calculus deals with the study of the rates at which quantities change. It is one of the two principal areas of calculus (integration being the other).
Start learning

A brief introduction to differential calculus

How would you like to follow in the footsteps of Euclid and Archimedes? Would you like to be able to determine precisely how fast Usain Bolt is accelerating exactly 2 seconds after the starting gun? Differential calculus deals with the study of the rates at which quantities change. It is one of the two principal areas of calculus (integration being the other).
Start learning
Watch an introduction video9 minutes 7 seconds

Unit 1: Limits and continuity

Limits intro: Limits and continuityEstimating limits from graphs: Limits and continuityEstimating limits from tables: Limits and continuityFormal definition of limits (epsilon-delta): Limits and continuityProperties of limits: Limits and continuityLimits by direct substitution: Limits and continuityLimits using algebraic manipulation: Limits and continuityStrategy in finding limits: Limits and continuity
Squeeze theorem: Limits and continuityTypes of discontinuities: Limits and continuityContinuity at a point: Limits and continuityContinuity over an interval: Limits and continuityRemoving discontinuities: Limits and continuityInfinite limits: Limits and continuityLimits at infinity: Limits and continuityIntermediate value theorem: Limits and continuity

Unit 2: Derivatives: definition and basic rules

Average vs. instantaneous rate of change: Derivatives: definition and basic rulesSecant lines: Derivatives: definition and basic rulesDerivative definition: Derivatives: definition and basic rulesEstimating derivatives: Derivatives: definition and basic rulesDifferentiability: Derivatives: definition and basic rulesPower rule: Derivatives: definition and basic rules
Derivative rules: constant, sum, difference, and constant multiple: Derivatives: definition and basic rulesCombining the power rule with other derivative rules: Derivatives: definition and basic rulesDerivatives of cos(x), sin(x), 𝑒ˣ, and ln(x): Derivatives: definition and basic rulesProduct rule: Derivatives: definition and basic rulesQuotient rule: Derivatives: definition and basic rulesDerivatives of tan(x), cot(x), sec(x), and csc(x): Derivatives: definition and basic rulesProof videos: Derivatives: definition and basic rules

Unit 3: Derivatives: chain rule and other advanced topics

Chain rule: Derivatives: chain rule and other advanced topicsMore chain rule practice: Derivatives: chain rule and other advanced topicsImplicit differentiation: Derivatives: chain rule and other advanced topicsImplicit differentiation (advanced examples): Derivatives: chain rule and other advanced topicsDifferentiating inverse functions: Derivatives: chain rule and other advanced topicsDerivatives of inverse trigonometric functions: Derivatives: chain rule and other advanced topics
Strategy in differentiating functions: Derivatives: chain rule and other advanced topicsDifferentiation using multiple rules: Derivatives: chain rule and other advanced topicsSecond derivatives: Derivatives: chain rule and other advanced topicsDisguised derivatives: Derivatives: chain rule and other advanced topicsLogarithmic differentiation: Derivatives: chain rule and other advanced topicsProof videos: Derivatives: chain rule and other advanced topics

Unit 4: Applications of derivatives

Meaning of the derivative in context: Applications of derivativesStraight-line motion: Applications of derivativesNon-motion applications of derivatives: Applications of derivativesIntroduction to related rates: Applications of derivatives
Solving related rates problems: Applications of derivativesApproximation with local linearity: Applications of derivativesL’Hôpital’s rule: Applications of derivativesL’Hôpital’s rule: composite exponential functions: Applications of derivatives

Unit 5: Analyzing functions

Mean value theorem: Analyzing functions Extreme value theorem and critical points: Analyzing functions Intervals on which a function is increasing or decreasing: Analyzing functions Relative (local) extrema: Analyzing functions Absolute (global) extrema: Analyzing functions Concavity and inflection points intro: Analyzing functions
Analyzing concavity and inflection points: Analyzing functions Second derivative test: Analyzing functions Sketching curves: Analyzing functions Connecting f, f', and f'': Analyzing functions Solving optimization problems: Analyzing functions Analyzing implicit relations: Analyzing functions Calculator-active practice: Analyzing functions

Unit 6: Parametric equations, polar coordinates, and vector-valued functions

Parametric equations intro: Parametric equations, polar coordinates, and vector-valued functionsSecond derivatives of parametric equations: Parametric equations, polar coordinates, and vector-valued functionsVector-valued functions: Parametric equations, polar coordinates, and vector-valued functions
Planar motion: Parametric equations, polar coordinates, and vector-valued functionsPolar functions: Parametric equations, polar coordinates, and vector-valued functions

Course challenge

Test your knowledge of the skills in this course. Have a test coming up? The Course challenge can help you understand what you need to review.
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