Unit 06 · Lesson 07 · Week 18 · Chain Rule, Implicit & Inverse Functions
Derivatives of Inverse and Inverse Trigonometric Functions
The derivative of an inverse reverses input and output while taking the reciprocal of the original slope at the corresponding point. A zero original slope prevents a finite inverse derivative. This lesson emphasizes derivatives of inverse and inverse trigonometric functions through symbolic, graphical, numerical, and contextual evidence.
Learning intentions
- Define and use the central calculus language in derivatives of inverse and inverse trigonometric functions.
- Move among symbolic, graphical, numerical, and contextual representations.
- Show complete reasoning with correct notation, domain conditions, units, and intervals.
- Verify a conclusion using a second method or representation.
Essential question
How do nested, implicit, and inverse relationships change the way derivatives are calculated and interpreted?
Visual model
The derivative of an inverse reverses input and output while taking the reciprocal of the original slope at the corresponding point. A zero original slope prevents a finite inverse derivative.
(f^{-1})'(a)=\frac1{f'(f^{-1}(a))}
Worked reasoning
Misconception check: The inverse derivative is not simply 1/f'(a); the original derivative must be evaluated at f^{-1}(a).
Three learning pathways
Foundation
Reconstruct definitions, annotate a worked example, and solve representative problems with guided prompts.
Core
Complete mixed representations, justify every major step, and verify exact and approximate results.
Extended
Generalize a result, analyze an error or edge case, and connect the lesson to modelling or proof-informed reasoning.
Communication and modelling
State what is known, identify the theorem or structure used, show symbolic work, interpret the result in context, and explain how the conclusion was checked. Calculator or CAD evidence must include settings, units, and a written interpretation.
Khaemenes Academy · Open-Age Calculus I
Derivatives of Inverse and Inverse Trigonometric Functions
Student Worksheet · Analyze, calculate, graph, or justify each result. Show reasoning, intervals, units, and technology evidence when used.
Name
Date
If f(2)=5 and f'(2)=4, find (f⁻¹)'(5).
Differentiate arcsin x.
Differentiate arctan x.
If f'(c)=0, what happens to the inverse derivative at f(c)?
Why is the input to f' equal to f⁻¹(a)?
Differentiate arccos x.
Lesson record
Mark this lesson complete only after the worksheet, correction, and reflection are finished.