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AP Calc AB · Chapter 3 of 8

Composite, Implicit, and Inverse Functions

Differentiate composite functions with the chain rule, handle implicit relationships, and compute derivatives of inverse functions.

Why this chapter matters

Many real formulas are nested or not solved for one variable, so these methods make derivatives practical in realistic settings.

What you will learn

  • Apply the chain rule to nested function structures.
  • Use implicit differentiation to find dy/dx when x and y are linked.
  • Find derivatives involving inverse functions and interpret restrictions.

Lessons in this chapter

  1. Chain rule for composite functionsTrack outer and inner functions and multiply derivative layers correctly. Read the full guide →
  2. Implicit differentiation workflowDifferentiate both sides with respect to x and isolate dy/dx. Read the full guide →
  3. Derivatives of inverse functionsUse inverse relationships to connect derivatives at paired points.
  4. Logarithmic differentiation basicsDifferentiate products and powers more efficiently by taking logs.

Study task

Differentiate y = (3x^2 + 1)^5 and x^2 + xy + y^2 = 7, then explain why the second problem requires implicit differentiation.

Chapter checkpoint

If y^3 + x^2y = 10, find dy/dx.

Differentiate implicitly: 3y^2(dy/dx) + 2xy + x^2(dy/dx) = 0. So dy/dx = -2xy / (3y^2 + x^2).