# Derivatives of Basic Functions

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## Lecture Notes

#### The Derivative of xn

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Section 3, Page 3

Derivative formula given for functions of the form f(x) = xn, derived using the binomial theorem.

Instructor: Prof. Jason Starr
Prior Knowledge: Concept of derivative (section 2 of lecture 1) and the binomial theorem (section 2 of this lecture)

#### Another Proof That d(xn)/dx = nx(n-1)

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Section 6, Page 4

Proof by induction of derivative formula for xn.

Instructor: Prof. Jason Starr
Prior Knowledge: Knowledge of mathematical induction, Product Rule (section 4 of this lecture), and concept of derivative (section 2 of lecture 1)

#### The Derivative of un

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Section 2, Page 1 to page 2

Derivative formula for un, proven by induction.

Instructor: Prof. Jason Starr
Prior Knowledge: Product Rule (section 1 of this lecture) and knowledge of mathematical induction

#### The Derivative of xa, a a Fraction

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Section 3, Page 2 to page 3

Derivative formula found for functions of the form f(x) = xa, where a is a fraction.

Instructor: Prof. Jason Starr
Prior Knowledge: Derivative of xn (sections 3 and 6 of lecture 3)

#### Rules for Exponentials and Logarithms

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Section 2, Page 1 to page 2

Algebraic rules for exponentials and logarithms are reviewed.

Instructor: Prof. Jason Starr
Prior Knowledge: None

#### The Derivative of ax

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Section 3, Page 2 to page 3

Derivation, leading to the definition of e and the natural logarithm.

Instructor: Prof. Jason Starr
Prior Knowledge: Rules for Exponentials (section 2 of this lecture)

#### The Derivative of log_a(x) and the Value of e

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Section 4, Page 3 to page 4

Derivation using the chain rule. Derivative of ln(x) also given and used to find the numeric value of e.

Instructor: Prof. Jason Starr
Prior Knowledge: Rules for Logarithms (section 2 of this lecture) and definition of derivative (section 2 of lecture 1)

#### 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.

Instructor: Prof. Jason Starr
Prior Knowledge: Rules for Logarithms (section 2 of this lecture) and Product Rule (section 4 of lecture 3)

#### Trigonometric Functions

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Section 1, Page 1 to page 2

Angles and continuous functions of them are described abstractly, with mention of the specific functions sin, cos, tan, sec, csc, and cot.

Instructor: Prof. Jason Starr
Prior Knowledge: None

#### Trigonometric Identities

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Section 2, Page 2

Angle addition formulas and other trigonometric identities involving sin and cos.

Instructor: Prof. Jason Starr
Prior Knowledge: None

#### Derivatives of sin(x) and cos(x)

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Section 4, Page 3

Derivation using trig identities and difference quotients.

Instructor: Prof. Jason Starr
Prior Knowledge: Trigonometric Identities (section 2 of this lecture) and difference quotients (section 2 of lecture 1)

#### Derivatives of Other Trigonometric Functions

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Section 5, Page 4

Derivative of tan(x) is derived from the quotient rule and the derivatives of sin(x) and cos(x). Derivatives for sec(x), csc(x), and cot(x) are also stated.

Instructor: Prof. Jason Starr
Prior Knowledge: Derivatives of sin(x) and cos(x)

#### The Inverse Trigonometric Functions

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Section 3, Page 2

Brief definitions of the inverse trigonometric functions sin-1(x), cos-1(x), and tan-1(x)

Instructor: Prof. Jason Starr
Prior Knowledge: Inverse Functions (section 1 of this lecture)

#### Derivatives of the Inverse Trigonometric Functions

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Section 5, Page 3 to page 4

Formulas for the derivatives of the inverse trigonometric functions, as well as the equation sin-1(x) + cos-1(x) = pi/2.

Instructor: Prof. Jason Starr
Prior Knowledge: Inverse Trigonometric Functions and Derivatives of Inverse Functions (sections 3 and 4 of this lecture)

## Online Textbook Chapters

#### The Exponential Function

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Definition, including the properties of the function and its derivatives, as well as a graph of the function.

Prior Knowledge: Functions (OT1.3)

#### Properties of Trigonometric Functions

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List of important properties, as well as the derivatives of sine and cosine and a power series representation of sine and cosine.

Instructor: Prof. Daniel J. Kleitman
Prior Knowledge: Trigonometric Functions (OT2.2)

#### Philosophic Implications

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Deriving further rules for derivatives, including the product rule and the rule for functions of the form xn.

Instructor: Prof. Daniel J. Kleitman
Prior Knowledge: Differentiability (OT6.1)

#### Derivatives of the Basic Functions

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Derivatives for the identity, exponential, and sine functions.

Instructor: Prof. Daniel J. Kleitman
Prior Knowledge: Differentiability (OT6.1)

## Practice Problems

#### Tangent Line to the Graph of an Exponential Function

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Problem 1 (page 1)

Finding the equation for the tangent line to an exponential function through a point not on the graph of the function.

Instructor: Prof. Jason Starr
Prior Knowledge: None

#### Derivatives of Exponential and Trigonometric Functions

PDF
Problem 4 (page 2 to page 3)

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

Instructor: Prof. Jason Starr
Prior Knowledge: None

## Exam Questions

#### First, Second, and Third Derivatives

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Problem 2 (page 3)

Finding the derivatives of an exponential function.

Instructor: Prof. Jason Starr
Prior Knowledge: None

#### Computing Derivatives

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Problem 6 (page 7)

Finding the derivatives of exponential and logarithmic functions.

Instructor: Prof. Jason Starr
Prior Knowledge: None

#### Derivatives of Trigonometric Functions

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Problem 1 (page 2)

Finding the derivatives of two trigonometric functions.

Instructor: Prof. Jason Starr
Prior Knowledge: None

#### Derivative of an Inverse Function

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Problem 1.5 (page 1) to problem (page 2)

Evaluating the derivative of the inverse of an exponential function.

Instructor: Prof. Jason Starr
Prior Knowledge: None

#### Evaluating Derivatives and Limits

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Problem 1 (page 1)

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

Instructor: Prof. David Jerison
Prior Knowledge: None

#### The Inverse Sine Function

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Problem 4 (page 1)

Sketching the graph of the inverse sine function and finding its derivative.

Instructor: Prof. David Jerison
Prior Knowledge: None

#### Evaluating Derivatives

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

Two questions finding the derivatives of functions.

Instructor: Prof. David Jerison
Prior Knowledge: None

#### Evaluating Derivatives

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Problem 2 (page 1)

Finding the derivatives of four functions.

Instructor: Prof. David Jerison
Prior Knowledge: None

#### Trigonometric Formulas and Identities

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Problem (page 1)

A list of trigonometric and inverse trigonometric identities and formulas involving integrals and derivatives.

Instructor: Prof. David Jerison
Prior Knowledge: None

#### Evaluating Derivatives

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Problem 1 (page 1)

Three derivatives to be evaluated using a variety of techniques.

Instructor: Prof. David Jerison
Prior Knowledge: None

#### Differentiation Formulas: Polynomials, Products, Quotients

PDF - 2.2 MB
Problem 1E-1 (page 4) to problem 1E-5 (page 5)

Five questions which involve taking derivatives and antiderivatives of polynomials, finding the points on a graph which have horizontal tangent lines, and finding derivatives of rational functions.

Instructor: Prof. David Jerison
Prior Knowledge: None

#### Higher Derivatives

PDF - 2.2 MB
Problem 1G-1 (page 5) to problem 1G-5 (page 6)

Five questions which involve finding second, third, or nth derivatives of functions.

Prior Knowledge: None

#### Exponentials and Logarithms: Calculus

PDF - 2.2 MB
Problem 1I-1 (page 8) to problem 1I-5 (page 8)

Five questions which involve evaluating derivatives and limits of functions which contain logarithms or exponentials, graphing an exponential function, and calculating interest compounded with different frequencies.

Instructor: Prof. David Jerison
Prior Knowledge: None

#### Trigonometric Functions

PDF - 2.2 MB
Problem 1J-1 (page 9) to problem 1J-4 (page 9)

Four questions which involve calculating derivatives of trigonometric functions.

Instructor: Prof. David Jerison
Prior Knowledge: None

#### Differentials and Indefinite Integration

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.

Instructor: Prof. David Jerison
Prior Knowledge: None