- The Derivative of x
- Another Proof That d(x
^{n})/dx = nx^{(n-1)} - The Derivative of u
^{n} - The Derivative of x
^{a},*a*a Fraction - Rules for Exponentials and Logarithms
- The Derivative of a
^{x} - The Derivative of log_a(x) and the Value of e
- Logarithmic Differentiation
- Trigonometric Functions
- Trigonometric Identities
- Derivatives of sin(x) and cos(x)
- Derivatives of Other Trigonometric Functions
- The Inverse Trigonometric Functions
- Derivatives of the Inverse Trigonometric Functions
- The Exponential Function
- Properties of Trigonometric Functions
- Philosophic Implications
- Derivatives of the Basic Functions

- Tangent Line to the Graph of an Exponential Function
- Derivatives of Exponential and Trigonometric Functions
- First, Second, and Third Derivatives
- Computing Derivatives
- Derivatives of Trigonometric Functions
- Derivative of an Inverse Function
- Evaluating Derivatives and Limits
- The Inverse Sine Function
- Evaluating Derivatives
- Trigonometric Formulas and Identities
- Differentiation Formulas: Polynomials, Products, Quotients
- Higher Derivatives
- Exponentials and Logarithms: Calculus
- Differentials and Indefinite Integration
- Inverse Trigonometric and Hyperbolic Functions

Derivative formula given for functions of the form f(x) = x^{n}, derived using the binomial theorem.

18.01

*Single Variable Calculus*, Fall 2005

Prof. Jason Starr

**Course Material Related to This Topic:**

Proof by induction of derivative formula for x^{n}.

18.01

*Single Variable Calculus*, Fall 2005

Prof. Jason Starr

**Course Material Related to This Topic:**

Derivative formula for u^{n}, proven by induction.

18.01

*Single Variable Calculus*, Fall 2005

Prof. Jason Starr

**Course Material Related to This Topic:**

Derivative formula found for functions of the form f(x) = x^{a}, where *a* is a fraction.

18.01

*Single Variable Calculus*, Fall 2005

Prof. Jason Starr

**Course Material Related to This Topic:**

Algebraic rules for exponentials and logarithms are reviewed.

18.01

*Single Variable Calculus*, Fall 2005

Prof. Jason Starr

**Course Material Related to This Topic:**

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

18.01

*Single Variable Calculus*, Fall 2005

Prof. Jason Starr

**Course Material Related to This Topic:**

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

18.01

*Single Variable Calculus*, Fall 2005

Prof. Jason Starr

**Course Material Related to This Topic:**

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

18.01

*Single Variable Calculus*, Fall 2005

Prof. Jason Starr

**Course Material Related to This Topic:**

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

18.01

*Single Variable Calculus*, Fall 2005

Prof. Jason Starr

**Course Material Related to This Topic:**

Four questions which involve calculating derivatives of trigonometric functions.

18.01

*Single Variable Calculus*, Fall 2006

Prof. David Jerison

**Course Material Related to This Topic:**

- Complete exam problems 1J–1 to 1J–4 on page 9
- Check solution to exam problems 1J–1 to 1J–4 on pages 14–6

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

18.01

*Single Variable Calculus*, Fall 2005

Prof. Jason Starr

**Course Material Related to This Topic:**

Derivation using trig identities and difference quotients.

18.01

*Single Variable Calculus*, Fall 2005

Prof. Jason Starr

**Course Material Related to This Topic:**

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.

18.01

*Single Variable Calculus*, Fall 2005

Prof. Jason Starr

**Course Material Related to This Topic:**

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

18.01

*Single Variable Calculus*, Fall 2005

Prof. Jason Starr

**Course Material Related to This Topic:**

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

18.01

*Single Variable Calculus*, Fall 2005

Prof. Jason Starr

**Course Material Related to This Topic:**

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

**Course Material Related to This Topic:**

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

18.013A

*Calculus with Applications*, Spring 2005

Prof. Daniel J. Kleitman

**Course Material Related to This Topic:**

Deriving further rules for derivatives, including the product rule and the rule for functions of the form x^{n}.

18.013A

*Calculus with Applications*, Spring 2005

Prof. Daniel J. Kleitman

**Course Material Related to This Topic:**

Derivatives for the identity, exponential, and sine functions.

18.013A

*Calculus with Applications*, Spring 2005

Prof. Daniel J. Kleitman

**Course Material Related to This Topic:**

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

18.01

*Single Variable Calculus*, Fall 2005

Prof. Jason Starr

**Course Material Related to This Topic:**

- Complete practice problem 1 on page 1
- Check solution to practice problem 1 on pages 6–7

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

18.01

*Single Variable Calculus*, Fall 2005

Prof. Jason Starr

**Course Material Related to This Topic:**

- Complete practice problem 4 on pages 2–3
- Check solution to practice problem 4 on pages 9–10

Finding the derivatives of an exponential function.

18.01

*Single Variable Calculus*, Fall 2005

Prof. Jason Starr

**Course Material Related to This Topic:**

- Complete exam problem 2 on page 3
- Check solution to exam problem 2 on pages 2–3

Finding the derivatives of an exponential function.

18.01

*Single Variable Calculus*, Fall 2005

Prof. Jason Starr

**Course Material Related to This Topic:**

- Complete exam problem 6 on page 7
- Check solution to exam problem 6 on pages 8–10

Finding the derivatives of two trigonometric functions.

18.01

*Single Variable Calculus*, Fall 2005

Prof. Jason Starr

**Course Material Related to This Topic:**

- Complete exam problem 1 on page 2
- Check solution to exam problem 1 on pages 1–2

Evaluating the derivative of the inverse of an exponential function.

18.01

*Single Variable Calculus*, Fall 2005

Prof. Jason Starr

**Course Material Related to This Topic:**

- Complete exam problem 1.5 on pages 1–2
- Check solution to exam problem 1.5 on pages 1–2

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

18.01

*Single Variable Calculus*, Fall 2006

Prof. David Jerison

**Course Material Related to This Topic:**

- Complete exam problem 1 on page 1
- Check solution to exam problem 1 on page 1

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

18.01

*Single Variable Calculus*, Fall 2006

Prof. David Jerison

**Course Material Related to This Topic:**

- Complete exam problem 4 on page 1
- Check solution to exam problem Question 4, page 1 of solution 1, pages 2–3 of solution 2

18.01

*Single Variable Calculus*, Fall 2006

Prof. David Jerison

**Course Material Related to This Topic:**

Two questions finding the derivatives of functions.

- Complete exam problems 1 to 2 on page 1
- Check solution to exam problems 1 to 2 on page 1

Finding the derivatives of four functions.

- Complete exam problem 2 on page 1
- Check solution to exam problem 2 on page 1

Three derivatives to be evaluated using a variety of techniques.

- Complete exam problem 1 on page 1
- Check solution to exam problem 1 on page 1

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

18.01

*Single Variable Calculus*, Fall 2006

Prof. David Jerison

**Course Material Related to This Topic:**

- Complete exam problem on page 1

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.

18.01

*Single Variable Calculus*, Fall 2006

Prof. David Jerison

**Course Material Related to This Topic:**

- Complete exam problems 1E–1 on page 4 to problems 1E–5 on page 5
- Check solution to exam problems on page 9

Five questions which involve finding second, third, or n^{th} derivatives of functions.

**Course Material Related to This Topic:**

- Complete exam problems 1G–1 on page 5 to problems 1G–5 on page 6
- Check solution to exam problems on pages 11–2

Five questions which involve finding second, third, or n^{th} derivatives of functions.

18.01

*Single Variable Calculus*, Fall 2006

Prof. David Jerison

**Course Material Related to This Topic:**

- Complete exam problems 1I–1 to 1I–5 on page 8
- Check solution to exam problems 1I–1 to 1I–5 on pages 13–4

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

18.01

*Single Variable Calculus*, Fall 2006

Prof. David Jerison

**Course Material Related to This Topic:**

- Complete exam problems 3A–1 to 3A–3 on page 21
- Check solution to exam problems 3A–1 to 3A–3 on pages 37–9

Six questions which involve evaluating integrals and derivatives of these functions, as well as graphing them and finding tangent lines or average values.

18.01

*Single Variable Calculus*, Fall 2006

Prof. David Jerison

**Course Material Related to This Topic:**

- Complete exam problems 5A–1 to 5A–6 on page 35
- Check solution to exam problems 5A–1 to 5A–6 on pages 69–71