Home ## The Six Pillars of CalculusThe Pillars: A Road MapA picture is worth 1000 words ## Trigonometry ReviewThe basic trig functionsBasic trig identities The unit circle Addition of angles, double and half angle formulas The law of sines and the law of cosines Graphs of Trig Functions ## Exponential FunctionsExponentials with positive integer exponentsFractional and negative powers The function $f(x)=a^x$ and its graph Exponential growth and decay ## Logarithms and Inverse functionsInverse FunctionsHow to find a formula for an inverse function Logarithms as Inverse Exponentials Inverse Trig Functions ## Intro to LimitsClose is good enoughDefinition One-sided Limits How can a limit fail to exist? Infinite Limits and Vertical Asymptotes Summary ## Limit Laws and ComputationsA summary of Limit LawsWhy do these laws work? Two limit theorems How to algebraically manipulate a 0/0? Limits with fractions Limits with Absolute Values Limits involving Rationalization Limits of Piece-wise Functions The Squeeze Theorem ## Continuity and the Intermediate Value TheoremDefinition of continuityContinuity and piece-wise functions Continuity properties Types of discontinuities The Intermediate Value Theorem Examples of continuous functions ## Limits at InfinityLimits at infinity and horizontal asymptotesLimits at infinity of rational functions Which functions grow the fastest? Vertical asymptotes (Redux) Toolbox of graphs ## Rates of ChangeTracking changeAverage and instantaneous velocity Instantaneous rate of change of any function Finding tangent line equations Definition of derivative ## The Derivative FunctionThe derivative functionSketching the graph of $f'$ Differentiability Notation and higher-order derivatives ## Basic Differentiation RulesThe Power Rule and other basic rulesThe derivative of $e^x$ ## Product and Quotient RulesThe Product RuleThe Quotient Rule ## Derivatives of Trig FunctionsTwo important LimitsSine and Cosine Tangent, Cotangent, Secant, and Cosecant Summary ## The Chain RuleTwo forms of the chain ruleVersion 1 Version 2 Why does it work? A hybrid chain rule ## Implicit DifferentiationIntroduction and ExamplesDerivatives of Inverse Trigs via Implicit Differentiation A Summary ## Derivatives of LogsFormulas and ExamplesLogarithmic Differentiation ## Derivatives in ScienceIn PhysicsIn Economics In Biology ## Related RatesOverviewHow to tackle the problems Example (ladder) Example (shadow) ## Linear Approximation and DifferentialsOverviewExamples An example with negative $dx$ ## Differentiation ReviewBasic Building BlocksAdvanced Building Blocks Product and Quotient Rules The Chain Rule Combining Rules Implicit Differentiation Logarithmic Differentiation Conclusions and Tidbits ## Absolute and Local ExtremaDefinitionsThe Extreme Value Theorem Fermat's Theorem How-to ## The Mean Value and other TheoremsRolle's TheoremsThe Mean Value Theorem Finding $c$ ## $f$ vs. $f'$Increasing/Decreasing Test and Critical NumbersHow-to The First Derivative Test Concavity, Points of Inflection, and the Second Derivative Test ## Indeterminate Forms and L'Hospital's RuleWhat does $\frac{0}{0}$ equal?Indeterminate Differences Indeterminate Powers Three Versions of L'Hospital's Rule Proofs ## OptimizationStrategiesAnother Example ## Newton's MethodThe Idea of Newton's MethodAn Example Solving Transcendental Equations When NM doesn't work ## Anti-derivativesAnti-derivatives and PhysicsSome formulas Anti-derivatives are not Integrals ## The Area under a curveThe Area Problem and ExamplesRiemann Sums Notation Summary ## Definite IntegralsDefinitionProperties What is integration good for? More Examples ## The Fundamental Theorem of CalculusThree Different QuantitiesThe Whole as Sum of Partial Changes The Indefinite Integral as Antiderivative The FTC and the Chain Rule |
## Combining RulesMany functions can't be cracked open with a single rule. Instead, we can use a rule to break the problem down into (possibly several) simpler problems. Then we can use another rule on each of those. And repeat as long as needed to get the answer.
There isn't always an obvious order in which to apply the rules. When in doubt, think about the structure of the function. Would you describe it as a product of two simpler functions? If so, apply the product rule first! Would you describe it as a quotient? If so, apply the quotient rule first. Is it instead a power, or a log, or an exponential, or a trig function of some complicated expression (which may itself involve products, quotients, or further nesting)? If so, apply the chain rule first. |