AP Calculus · Unit 3
Composite, Implicit and Inverse Functions
The chain rule, implicit differentiation and inverse derivatives.
Topic 3.1Differentiate a composition
- Identify the layers of a compositionDistinguish the inner input from the outer function.
- Apply the chain rule completelyDifferentiate the outer function at the inner input and include every inner derivative.
- Use the chain rule with a tableLook up outer derivatives at the transformed input.
- Use a composite derivative in contextFind a slope or rate for linked quantities with the correct units.
Topic 3.2Differentiate an implicit relation
- Differentiate an implicit relationTreat y as a function of x and include its derivative in each relevant term.
- Solve for the derivativeIsolate dy/dx, including product terms involving both variables.
- Find the tangent at a pointCheck the point lies on the curve and calculate its tangent slope or line.
- Find horizontal and vertical tangentsSolve the appropriate numerator or denominator condition and check exceptions.
Topic 3.3Differentiate an inverse function
- Find the matching inverse inputUse f(a)=b to connect the inverse's input b with the original input a.
- Calculate an inverse derivativeTake the reciprocal of the original derivative at the matching point.
- Use inverse information from dataRead the necessary value and slope from a table or graph.
- Check when the inverse rule appliesCheck the branch, local invertibility, and a nonzero original derivative.
Topic 3.4Differentiate inverse trigonometric functions
- Differentiate inverse sine and inverse cosineUse the correct square-root denominator and sign.
- Differentiate inverse tangentUse the reciprocal of one plus the squared input.
- Apply the chain rule inside an inverse trig functionInclude the inner derivative and evaluate at the requested input.
- Check where the derivative is definedDistinguish the inverse function's domain from its derivative's domain.
Topic 3.5Choose and combine derivative rules
- Choose the outermost derivative ruleRecognize whether the expression begins as a sum, product, quotient, or composition.
- Combine rules in one derivativeApply product or quotient rules together with the chain rule.
- Use combined rules with a tableMatch each function value and derivative to the correct input.
- Find and repair a rule errorIdentify a missing factor, wrong sign, or incorrectly differentiated term.
Topic 3.6Differentiate repeatedly
- Find a second derivativeDifferentiate the entire first derivative again.
- Use higher-derivative notationDistinguish derivative order from an exponent and evaluate the requested derivative.
- Interpret successive ratesConnect position, velocity, and acceleration with appropriate units.
- Recognize a repeating derivative patternUse trig or exponential patterns to find a higher derivative.