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Rules: Derivative of Sums, Products & Quotients of Functions


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Lecture Notes

Linearity

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Section 3, Page 3

Linearity of the derivative (derivative of sums of functions).

Course: 18.01 Single Variable Calculus, Fall 2005
Instructor: Prof. Jason Starr
Prior Knowledge: Concept of Derivative (section 2 of lecture 1)
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The Liebniz Rule / Product Rule

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Section 4, Page 3

Product rule for derivatives. Gives proof sketch.

Course: 18.01 Single Variable Calculus, Fall 2005
Instructor: Prof. Jason Starr
Prior Knowledge: Concept of Derivative (section 2 of lecture 1)
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The Quotient Rule

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Section 5, Page 3 to page 4

Quotient rule for derivatives, derived using product rule.

Course: 18.01 Single Variable Calculus, Fall 2005
Instructor: Prof. Jason Starr
Prior Knowledge: Product Rule (section 4 of this lecture) and Concept of Derivative (section 2 of lecture 1)
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Product Rule Example

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Section 1, Page 1

Product rule used to find a derivative.

Course: 18.01 Single Variable Calculus, Fall 2005
Instructor: Prof. Jason Starr
Prior Knowledge: Product Rule (section 4 of lecture 3) and derivative of xn (sections 3 and 6 of lecture 3)
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Logarithmic Differentiation

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Section 5, Page 4 to page 5

Finding the derivative of a product of functions using logarithms to convert into a sum of functions. Includes worked example.

Course: 18.01 Single Variable Calculus, Fall 2005
Instructor: Prof. Jason Starr
Prior Knowledge: Rules for Logarithms (section 2 of this lecture) and Product Rule (section 4 of lecture 3)
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Differentials

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Section 1, Page 1 to page 2

Differential notation for derivatives is explained and common rules for derivatives are listed in differential notation. Includes an example of a derivative taken in differential notation.

Course: 18.01 Single Variable Calculus, Fall 2005
Instructor: Prof. Jason Starr
Prior Knowledge: Concept of Derivative (section 2 of lecture 1)
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Online Textbook Chapter

Derivatives of Combinations of Functions

Document Document

Rules for the derivatives of sums and products of functions, as well as the chain rule and rules for finding the derivative of an inverse function.

Course: 18.013A Calculus with Applications, Spring 2005
Instructor: Prof. Daniel J. Kleitman
Prior Knowledge: Differentiability (OT6.1)
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Practice Problem

Derivatives of Exponential and Trigonometric Functions

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Problem 4 (page 2 to page 3)

Taking the first and second derivatives of a function involving an exponential and a cosine.

Course: 18.01 Single Variable Calculus, Fall 2005
Instructor: Prof. Jason Starr
Prior Knowledge: None

Solution (PDF) Pages 9 to 10.

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Exam Questions

Evaluating Derivatives and Limits

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Problem 1 (page 1)

Four-part question involving the evaluation of three derivatives and a limit.

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

Solution (PDF)# Page 1

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Evaluating Derivatives

Document PDF
Problem 1 (page 1) to problem 2 (page 1)

Two questions finding the derivatives of functions.

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

Solution (PDF)# Page 1

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Evaluating Derivatives

Document PDF
Problem 2 (page 1)

Finding the derivatives of four functions.

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

Solution (PDF)# Page 1

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Evaluating Derivatives

Document PDF
Problem 1 (page 1)

Three derivatives to be evaluated using a variety of techniques.

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

Solution (PDF)# Page 1

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Differentials and Indefinite Integration

Document PDF - 2.2 MB
Problem 3A-1 (page 21) to problem 3A-3 (page 21)

Three questions which involve evaluating five differentials and twenty indefinite integrals using a range of techniques.

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

Solution (PDF - 4.0 MB) Pages 37 to 39.

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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