18.01SC | Fall 2010 | Undergraduate

Single Variable Calculus

1. Differentiation

Part B: Implicit Differentiation and Inverse Functions

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This section extends the methods of Part A to exponential and implicitly defined functions. By the end of Part B, we are able to differentiate most elementary functions.

» Session 13: Implicit Differentiation
» Session 14: Examples of Implicit Differentiation
» Session 15: Implicit Differentiation and Inverse Functions
» Session 16: The Derivative of a{{< sup “x” >}}
» Session 17: The Exponential Function, its Derivative, and its Inverse
» Session 18: Derivatives of other Exponential Functions
» Session 19: An Interesting Limit Involving e
» Session 20: Hyperbolic Trig Functions
» Problem Set 2

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Overview

In this session you will:

  • Do practice problems
  • Use the solutions to check your work

Problem Set

Use Differentiation (PDF) to do the problems below.

Section Topic Exercises
1F Chain rule, implicit differentiation 3, 5, 8a, 8c
1G Higher derivatives 4, 5b

Use Integration Techniques (PDF) to do the problems below.

Section Topic Exercises
5A Inverse trigonometric functions; Hyperbolic functions 1a, 1b, 1c (just sin, cos, sec), 3f, 3g, 3h

Use Differentiation (PDF) to do the problems below.

Section Topic Exercises
1H Exponentials and Logarithms: Algebra 1a, 1b, 2, 3a, 5b
1I Exponentials and Logarithms: Calculus 1c, 1d, 1e, 1f, 1m, 4a

Use Integration Techniques (PDF) to do the problems below.

Section Topic Exercises
5A Inverse trigonometric functions; Hyperbolic functions 5a, 5b, 5c

Solutions

Solutions to Differentiation problems (PDF)

Solutions to Integration Techniques problems (PDF)

This problem set is from exercises and solutions written by David Jerison and Arthur Mattuck.

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Overview

Up to now, we’ve been finding derivatives of functions. Implicit differentiation allows us to determine the rate of change of values that aren’t expressed as functions.

Lecture Video and Notes

Video Excerpts

Clip 1: Slope of Tangent to Circle: Direct

Clip 2: Slope of Tangent to Circle: Implicit

Clip 3: Example: y4+xy2-2=0

Recitation Video

Implicit Differentiation

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Worked Example

Implicit Differentiation and the Second

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Overview

An important application of implicit differentiation is to finding the derivatives of inverse functions. Here we find a formula for the derivative of an inverse, then apply it to get the derivatives of inverse trigonometric functions.

Lecture Video and Notes

Video Excerpts

Clip 1: Derivative of the Inverse of a Function

Clip 2: Derivative of the Arctan Function

Recitation Video

Graphing the Arctan Function

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Lecture Video and Notes

Video Excerpts

Clip 1: Derivative of the Arcsin Function

Recitation Video

Derivative of the Arccos Function

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Worked Example

Derivative of the Square Root Function

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Overview

In this session we define the exponential and natural log functions. We then use the chain rule and the exponential function to find the derivative of a^x.

Lecture Video and Notes

Video Excerpts

Clip 1: Definition of ex

Clip 2: Natural Log

Worked Example

Solving Equations with e and ln

Lecture Video and Notes

Video Excerpts

Clip 1: Derivative of ax

Recitation Video

Log and Exponent Derivatives

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Lecture Video and Notes

Video Excerpts

Clip 1: The Most Natural Logarithmic Function

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Overview

This session introduces the technique of logarithmic differentiation and uses it to find the derivative of a^x. Substituting different values for a yields formulas for the derivatives of several important functions. Further applications of logarithmic differentiation include verifying the formula for the derivative of x^r, where r is any real number.

Lecture Video and Notes

Video Excerpts

Clip 1: Other Bases

Clip 2: Derivative of ax, Logarithmic

Clip 3: Derivative of xx

Recitation Video

Rules of Logs

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Lecture Video and Notes

Video Excerpts

Clip 1: The Power Rule

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Overview

Hyperbolic trig functions are analogous to the trig functions like sine, cosine and tangent that we are already familiar with. They are used in mathematics, engineering and physics.

Reading

Hyperbolic Trig Functions (PDF)

Recitation Video

Hyperbolic Trig Functions

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Worked Example

sinh (x+y) and cosh (x+y) in Terms of Their Constituent Parts

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Course Info

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Fall 2010
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Undergraduate
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