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Unit 3: Differentiation: Composite, Implicit, and Inverse Functions

6 dot points across 6 inquiry questions. Click any dot point for a focused answer with worked past exam questions where available.

How do you differentiate a function built by composing one function inside another?

Topic 3.1 The Chain Rule: differentiate composite functions using the chain rule.
A focused answer to AP Calculus AB Topic 3.1, stating and applying the chain rule for composite functions, in both the Leibniz and outside-inside forms, with worked examples combining it with the power, trig, exponential and log rules.
9 min answer →

If you know how fast a function changes, how fast does its inverse change, without ever finding a formula for the inverse?

Topic 3.3 Differentiating Inverse Functions: find the derivative of the inverse of a function at a point using the reciprocal relationship.
A focused answer to AP Calculus AB Topic 3.3, deriving and applying the inverse-function derivative formula, which relates the slope of an inverse function to the reciprocal of the original function's slope, with worked point evaluations.
9 min answer →

What are the derivatives of arcsin, arctan and the other inverse trigonometric functions, and where do those algebraic expressions come from?

Topic 3.4 Differentiating Inverse Trigonometric Functions: state and apply the derivatives of the inverse trigonometric functions.
A focused answer to AP Calculus AB Topic 3.4, giving the derivatives of arcsin, arccos, arctan and the other inverse trig functions, showing where they come from, and combining them with the chain rule in worked examples.
9 min answer →

What does it mean to differentiate a derivative, and what does the second derivative tell you?

Topic 3.6 Calculating Higher-Order Derivatives: find second and higher-order derivatives and interpret their notation.
A focused answer to AP Calculus AB Topic 3.6, on second and higher-order derivatives, their notation, how to compute them by differentiating repeatedly, and what the second derivative means physically, with worked examples.
9 min answer →

How do you find a slope when y is not isolated, but tangled together with x in an equation?

Topic 3.2 Implicit Differentiation: find the derivative of a relation defined implicitly by an equation in x and y.
A focused answer to AP Calculus AB Topic 3.2, explaining implicit differentiation for relations where y is not solved for, treating y as a function of x and applying the chain rule, with worked examples and tangent-line problems.
9 min answer →

Faced with a complicated function, how do you decide which differentiation rules to use, and in what order?

Topic 3.5 Selecting Procedures for Calculating Derivatives: choose and combine the appropriate differentiation rules for a given function.
A focused answer to AP Calculus AB Topic 3.5, on recognizing the structure of a function and choosing which differentiation rules to apply and in what order, combining the power, product, quotient and chain rules, with worked multi-rule examples.
9 min answer →