Essay on Calculus: Derivative and Rule

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Calculus Study Guide
Table of Contents
Derivatives ………………………………………………………………… 2
Limits ...…………………………...……………………………………..… 23
Riemann Sums……………..……………………..…………………………28
Integrals…………………………………………..…………………………32

©2008 Rose-Hulman Institute of Technology’s Homework Hotline
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Derivatives
The derivative of a function is the instantaneous rate of change, or the slope of that function at a particular point. For example, the green line in the picture below is tangent to the curve ݂ሺ‫ݔ‬ሻ at the point ‫ ݔ‬ൌ .75. The slope of the green line in this picture is the derivative of ݂ሺ‫ݔ‬ሻ at the point ‫ ݔ‬ൌ 0.75.
Notations for the derivative include ݂′ሺ‫ݔ‬ሻ,

ௗ
ௗ௫

൫݂ሺ‫ݔ‬ሻ൯ ,

ௗ௬
ௗ௫

, and ‫ ݕ‬ᇱ .

݂ሺ‫ݔ‬ሻ

Slope= ݂’ሺ‫ݔ‬ሻ

We can define the derivative in terms of limits. Suppose we take two points on either side of a
.
point ‫ ,ݔ‬and draw the line between them, called a secant line. The slope of the secant line is
,
approximately the slope of the function at the point ‫.ݔ‬

࢞

࢞

Now move those points closer to ‫ ,ݔ‬and draw another secant line. The slope of this line is much
,
closer to the slope of the function at ‫ .ݔ‬If we take the limit of the slope of the secant line as these points move closer and closer together, we get the definition of the derivative at ‫.ݔ‬

©2008 Rose-Hulman Institute of Technology’s Homework Hotline
Hulman
Page 2 of 37

The Limit Definition
݂ሺ‫ݔ∆ + ݔ‬ሻ − ݂ሺ‫ݔ‬ሻ
∆௫→଴
∆‫ݔ‬

݂ ᇱ ሺ‫ݔ‬ሻ ൌ lim

The limit definition is the most basic formula for calculating the derivative of a function. In this formula, ݂ሺ‫ݔ‬ሻ is the function to be differentiated, and ∆‫ ݔ‬represents a small change in ‫ ,ݔ‬which goes to zero.

Example:
Find the derivative of ݂ሺ‫ݔ‬ሻ = ‫ ݔ‬ଶ . Then calculate the slope of ‫ ݔ‬ଶ at ‫.4 = ݔ‬
݂ ᇱ ሺ‫ݔ‬ሻ = lim∆௫→଴

௙ሺ௫ା∆௫ሻି௙ሺ௫ሻ

ሺ‫ݔ∆ + ݔ‬ሻଶ − ‫ ݔ‬ଶ
∆௫→଴
∆‫ݔ‬

݂ ᇱ ሺ‫ݔ‬ሻ = lim
݂

ᇱ ሺ‫ݔ‬ሻ

∆௫

‫ ݔ‬ଶ + 2‫ + ݔ∆ ∙ ݔ‬ሺ∆‫ݔ‬ሻଶ − ‫ ݔ‬ଶ
= lim
∆௫→଴
∆‫ݔ‬
2‫ + ݔ∆ ∙ ݔ‬ሺ∆‫ݔ‬ሻଶ
∆௫→଴
∆‫ݔ‬

݂ ᇱ ሺ‫ݔ‬ሻ = lim

݂ ᇱ ሺ‫ݔ‬ሻ = lim 2‫ݔ∆ + ݔ‬
∆௫→଴

ࢌᇱ ሺ࢞ሻ = ૛࢞

Change ݂ሺ‫ݔ‬ሻ to ‫ ݔ‬ଶ .
FOIL the ሺ‫ݔ∆ + ݔ‬ሻଶ .
The x ଶ terms cancel.
The ∆‫ ݔ‬ᇱ ‫ ݏ‬cancel.

∆‫ ݔ‬goes to zero.

To find the slope of ݂ሺ‫ݔ‬ሻ at the point ‫ ,4 = ݔ‬we plug in 4 to the derivative function.

݂ ᇱ ሺ‫ݔ‬ሻ = 2‫ݔ‬
݂ ᇱ ሺ4ሻ = 2ሺ4ሻ = 8

The slope of the line that is tangent to ݂ሺ‫ݔ‬ሻ when ‫ 4 = ݔ‬is 8.
Sometimes, it is not possible to find the derivative of a function at a certain point because that function is not continuous.

©2008 Rose-Hulman Institute of Technology’s Homework Hotline
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Example: Find the derivative of |‫ |ݔ‬at ‫.0 = ݔ‬
We can do this one using the limit definition (try it for practice!), or we can look at it graphically: ݂ሺ‫ ݔ‬ሻ = |‫| ݔ‬

Slope = 1

Slope = -1

The absolute value function has a derivative of ±1, depending on the sign of ‫.ݔ‬
1 ݂‫0 > ݔ ݎ݋‬
݂ ᇱ ሺ‫ݔ‬ሻ = ൜
−1 ݂‫0 < ݔ ݎ݋‬

But what is the derivative at ‫?0 = ݔ‬
When you studied limits, you probably learned that when the right-hand and left-hand limits at a point don’t agree, the limit at that point does not exist.
The derivative of |‫ |ݔ‬does not exist at ‫ 0 = ݔ‬because the function is not continuous there.

Rules for Finding Derivatives
In many cases, the limit definition of the derivative can be bypassed. The following differentiation rules show some of the shortcuts that can be taken.
The Power Rule
The power rule is one of the methods of taking the derivative that you will use most often, so you should make yourself familiar with it.
For any function ܽ ∙ ‫ ݔ‬௡ , the derivative can be found as:

©2008 Rose-Hulman Institute of Technology’s Homework Hotline
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ௗ

ௗ௫

ܽ ∙ ‫ ݔ‬௡ = ܽ ∙ ݊ ∙ ‫ ݔ‬௡ିଵ

Examples:

ௗ

ௗ௫
ௗ

ௗ௫
ௗ

ௗ௫

To apply this rule, multiply the coefficient, ܽ, by the current exponent, ݊, and then decrease the exponent on ‫ݔ‬ by 1.
Multiply by 2, then decrease the exponent to 1.

ሺ‫
Show More

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