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.

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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

### 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

### Lecture Video and Notes

#### Video Excerpts

Clip 1: Derivative of the Arcsin Function

### Worked Example

Derivative of the Square Root Function

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### Overview

We know how to take the derivative of a variable raised to a constant power. In this session we ask how to take the derivative of a constant raised to a variable power!

### Lecture Video and Notes

#### Video Excerpts

Clip 1: Differentiating Logs and Exponentials

Clip 2: Exponent Review

### Worked Example

Compound Interest

### Lecture Video and Notes

#### Video Excerpts

Clip 1: Difference Quotient of ax

Clip 2: Slope of ax at 0

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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

### 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

### Lecture Video and Notes

#### Video Excerpts

Clip 1: The Power Rule

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### Overview

We’ve defined the function e^x and we know its derivative. What can we say about the number e?

### Lecture Video and Notes

#### Video Excerpts

Clip 1: Value of e

Clip 2: Value of e, Revisited

### Worked Examples

Evaluating an Interesting Limit

Evaluating Interest Using the Limit

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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.

Hyperbolic Trig Functions (PDF)

### Worked Example

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

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## Course Info

Fall 2010
##### Learning Resource Types
Exams with Solutions
Lecture Notes
Lecture Videos
Problem Sets with Solutions
Simulations
Recitation Videos