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Implicit Differentiation & Derivatives of Inverse Functions


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Lecture Notes

Inverse Functions

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Section 1, Page 1 to page 2

Definition of inverse and invertible functions, including the necessary conditions for a function to be invertible.

Course: 18.01 Single Variable Calculus, Fall 2005
Instructor: Prof. Jason Starr
Prior Knowledge: None
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The Graph of an Inverse Function

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Section 2, Page 2

The graph of the inverse of y = f(x) is the graph of x = f(y) (switch the roles of x and y or reflect across the line y = x).

Course: 18.01 Single Variable Calculus, Fall 2005
Instructor: Prof. Jason Starr
Prior Knowledge: Inverse Functions (section 1 of this lecture)
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Derivatives of Inverse Functions

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Section 4, Page 2 to page 3

Formula for the derivative of an inverse function.

Course: 18.01 Single Variable Calculus, Fall 2005
Instructor: Prof. Jason Starr
Prior Knowledge: Inverse Functions (section 1 of this lecture)
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Exam Question

Implicit Differentiation: Cosine Function

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Problem 3 (page 1)

Finding the derivative for inverse cosine using the derivative for cosine and the method of implicit differentiation.

Course: 18.01 Single Variable Calculus, Fall 2006
Instructor: Prof. David Jerison
Prior Knowledge: None

Solution (PDF)# Page 1

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Analysis of Graphs Limits of Functions Asymptotic & Unbounded Behavior Continuity: Property of Functions Parametric, Polar & Vector Function Concept of the Derivative Derivative at a Point Derivative as a Function Second Derivatives Applications of Derivatives Computation of Derivatives Interpretations & Properties of Definite Integrals Applications of Integrals Fundamental Theorem of Calculus Applications of Antidifferentiation Numerical Approximations to Definite Integ Concept of Series Series of Constants Taylor Series Graphs of Functions Intuitive Understanding: Limiting Process Calculating Limits Using Algebra Asymptotic Behavior: Infinity Continuity Continuity in Terms of Limits Analysis of Planar Curves Derivative Presented Graphically, Numerically & Analytically Derivative Interpreted as Instantaneous Rate of Change Derivative: Limit of the Difference Quotient Differentiability & Continuity Tangent Line - Curve at a Point & Local Linear Approximation Instantaneous Rate of Change Approximate Rate of Change Characteristics of f & f Relationship Between Behavior of f & Sign of f Mean Value Theorem & Geometric Consequences Characteristics: Graphs of f, f & f Relationship Between Concavity of f & Sign of f'' Points of Inflection - Concavity Changes Analysis of Curves Applications of Derivatives Optimization: Absolute & Relative Extrema Modeling Rates of Change Implicit Differentiation & Derivatives of Inverse Functions Derivative as a Rate of Change l'Hôpital's Rule Geometric Interpretation of Differential Equations Derivatives of Basic Functions Rules: Derivative of Sums, Products & Quotients of Functions Chain Rule & Implicit Differentiation Derivatives: Parametric, Polar & Vector Functions Definite Integral - Limit of Riemann Sums Rate of Change of a Quantity/Change of Quantity over Interval Properties of Definite Integrals Appropriate Integrals Evaluate Definite Integrals Representation of Specific Antiderivatives Antiderivatives From Derivatives of Basic Functions Integration by Substitution, Parts & Partial Fractions Improper Integrals Separable Equations & Modeling Riemann & Trapezoidal Sums Series, Convergence, Divergence Motivating Examples Geometric Series with Applications The Harmonic Series Alternating Series - Error Bound Series as Riemann Sums & the Integral Test Ratio Test - Convergence/Divergence Comparison Test Taylor Polynomial Approximation Maclaurin & Taylor Series Maclaurin Series for Basic Functions Manipulating Taylor Series Power Series Defining Functions Radius/Interval of Convergence Lagrange Error Bound