Derivatives of Basic Functions

The Derivative of x

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

• 18.01 Single Variable Calculus, Fall 2005
  Prof. Jason Starr

Course Material Related to This Topic:

• Read lecture notes, section 3 on page 3

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

Proof by induction of derivative formula for xⁿ.

• 18.01 Single Variable Calculus, Fall 2005
  Prof. Jason Starr

Course Material Related to This Topic:

• Read lecture notes, section 6 on page 4

The Derivative of uⁿ

Derivative formula for uⁿ, proven by induction.

• 18.01 Single Variable Calculus, Fall 2005
  Prof. Jason Starr

Course Material Related to This Topic:

• Read lecture notes, section 2 on pages 1–2

The Derivative of x^a, a a Fraction

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:

• Read lecture notes, section 3 on pages 2–3

Rules for Exponentials and Logarithms

Algebraic rules for exponentials and logarithms are reviewed.

• 18.01 Single Variable Calculus, Fall 2005
  Prof. Jason Starr

Course Material Related to This Topic:

• Read lecture notes, section 2 on pages 1–2

The Derivative of a^x

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:

• Read lecture notes, section 3 on pages 2–3

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

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:

• Read lecture notes, section 4 on pages 3–4

Logarithmic Differentiation

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:

• Read lecture notes, section 4 on pages 4–5

Trigonometric Functions

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:

• Read lecture notes, section 1 on pages 1–2

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

Trigonometric Identities

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:

• Read lecture notes, section 2 on page 2

Derivatives of sin(x) and cos(x)

Derivation using trig identities and difference quotients.

• 18.01 Single Variable Calculus, Fall 2005
  Prof. Jason Starr

Course Material Related to This Topic:

• Read lecture notes, section 4 on page 3

Derivatives of Other Trigonometric Functions

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:

• Read lecture notes, section 5 on page 4

The Inverse Trigonometric Functions

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:

• Read lecture notes, section 3 on page 2

Derivatives of the Inverse Trigonometric Functions

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:

• Read lecture notes, section 5 on pages 3–4

The Exponential Function

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

Course Material Related to This Topic:

• Read chapter 2 of online textbook

Properties of Trigonometric Functions

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:

• Read chapter 2 of online textbook

Philosophic Implications

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

• 18.013A Calculus with Applications, Spring 2005
  Prof. Daniel J. Kleitman

Course Material Related to This Topic:

• Read chapter 6 of online textbook

Derivatives of the Basic Functions

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:

• Read chapter 8 of online textbook

Tangent Line to the Graph of an Exponential Function

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

Derivatives of Exponential and Trigonometric Functions

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

First, Second, and Third Derivatives

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

Computing Derivatives

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

Derivatives of Trigonometric Functions

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

Derivative of an Inverse Function

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

Evaluating Derivatives and Limits

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

The Inverse Sine Function

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

Evaluating Derivatives

• 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

Trigonometric Formulas and Identities

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

Differentiation Formulas: Polynomials, Products, Quotients

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

Higher Derivatives

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

Exponentials and Logarithms: Calculus

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

Differentials and Indefinite Integration

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

Inverse Trigonometric and Hyperbolic Functions

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