## 17Calculus Derivatives - Using Derivatives To Analyze and Graph Functions

### Derivatives

Graphing

Related Rates

Optimization

Other Applications

### Practice

Calculus 1 Practice

Calculus 2 Practice

Practice Exams

Calculus Tools

### Articles

We can use derivatives to help when graphing a function. But wait, don't we have calculators and computers for that? Yes, we do. But calculators are very imprecise when graphing and really just give us a feel for what the function looks like. And, although computer graphing is more precise than calculators, they still don't usually give us exact values and are really just used to show generally what is going on. Once you learn how to use derivatives to describe a graph and how to interpret the results, you will have a much better idea of what is going on and how to use the graph for your application.

First and Second Derivatives

The first derivative and the second derivative give you information about the shape of the graph. Each derivative tells you different things but they do parallel one another, i.e. the pattern with the first derivative is repeated in the second derivative, as shown in this table. These may not make sense initially but, after you have studied each, they will.

First Derivative

1. increasing/decreasing intervals

2. critical points    $$f'(x)=0$$

3. first derivative test

Second Derivative

4. concavity

5. inflection points    $$f''(x)=0$$

6. second derivative test

For the discussion of a topic, select a link above, starting with the the first derivative, increasing and decreasing intervals. The topics need to be discussed in order, since they build on one another. Once you have studied each of these topics, come back to this page and we will help you put all your newly-learned knowledge together.

Asymptotes

We touched on asymptotes on the continuity page. After you are comfortable with that material, come back here and watch this next video. It has lots of good examples of specific types of vertical asymptotes.

### Krista King Math - How To Find Vertical Asymptotes [9min-56secs]

video by Krista King Math

Putting It All Together

Okay, at this point you should have read all six topics listed in the table above and worked those practice problems. These practice problems put all those concepts together to build a picture of what a graph should look like and how it behaves. In addition, you will need to have covered specifics about domain and range.

Practice

Unless otherwise instructed, use the first and second derivative as well as other concepts linked to on this page to sketch these functions. Show your work by building tables and label your graph carefully and completely.

Basic

$$\displaystyle{y=\frac{x-1}{x^2}}$$

Problem Statement

Use the first and second derivative to sketch the function $$\displaystyle{y=\frac{x-1}{x^2}}$$. Show your work by building tables and label your graph carefully and completely.

Solution

This problem is solved in two videos.

### 1355 video

video by PatrickJMT

### 1355 video

video by PatrickJMT

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$$\displaystyle{f(x)=\frac{x}{x+4}}$$

Problem Statement

Use the first and second derivative to sketch the function $$\displaystyle{f(x)=\frac{x}{x+4}}$$. Show your work by building tables and label your graph carefully and completely.

Solution

### 1357 video

video by PatrickJMT

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$$f(x)=3x^2-6x+5$$

Problem Statement

Use the first and second derivative to sketch the function $$f(x)=3x^2-6x+5$$. Show your work by building tables and label your graph carefully and completely.

Solution

### 1926 video

video by Krista King Math

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Intermediate

$$\displaystyle{f(x)=\frac{x}{\sqrt{x^2+1}}}$$

Problem Statement

Use the first and second derivative to sketch the function $$\displaystyle{f(x)=\frac{x}{\sqrt{x^2+1}}}$$. Show your work by building tables and label your graph carefully and completely.

Solution

This problem is solved in four videos.

### 1356 video

video by PatrickJMT

### 1356 video

video by PatrickJMT

### 1356 video

video by PatrickJMT

### 1356 video

video by PatrickJMT

Log in to rate this practice problem and to see it's current rating.

You CAN Ace Calculus

### Topics You Need To Understand For This Page

 domain and range asymptotes derivatives

### Trig Formulas

The Unit Circle

The Unit Circle [wikipedia]

Basic Trig Identities

Set 1 - basic identities

$$\displaystyle{ \tan(t) = \frac{\sin(t)}{\cos(t)} }$$

$$\displaystyle{ \cot(t) = \frac{\cos(t)}{\sin(t)} }$$

$$\displaystyle{ \sec(t) = \frac{1}{\cos(t)} }$$

$$\displaystyle{ \csc(t) = \frac{1}{\sin(t)} }$$

Set 2 - squared identities

$$\sin^2t + \cos^2t = 1$$

$$1 + \tan^2t = \sec^2t$$

$$1 + \cot^2t = \csc^2t$$

Set 3 - double-angle formulas

$$\sin(2t) = 2\sin(t)\cos(t)$$

$$\displaystyle{ \cos(2t) = \cos^2(t) - \sin^2(t) }$$

Set 4 - half-angle formulas

$$\displaystyle{ \sin^2(t) = \frac{1-\cos(2t)}{2} }$$

$$\displaystyle{ \cos^2(t) = \frac{1+\cos(2t)}{2} }$$

Trig Derivatives

 $$\displaystyle{ \frac{d[\sin(t)]}{dt} = \cos(t) }$$ $$\displaystyle{ \frac{d[\cos(t)]}{dt} = -\sin(t) }$$ $$\displaystyle{ \frac{d[\tan(t)]}{dt} = \sec^2(t) }$$ $$\displaystyle{ \frac{d[\cot(t)]}{dt} = -\csc^2(t) }$$ $$\displaystyle{ \frac{d[\sec(t)]}{dt} = \sec(t)\tan(t) }$$ $$\displaystyle{ \frac{d[\csc(t)]}{dt} = -\csc(t)\cot(t) }$$

Inverse Trig Derivatives

 $$\displaystyle{ \frac{d[\arcsin(t)]}{dt} = \frac{1}{\sqrt{1-t^2}} }$$ $$\displaystyle{ \frac{d[\arccos(t)]}{dt} = -\frac{1}{\sqrt{1-t^2}} }$$ $$\displaystyle{ \frac{d[\arctan(t)]}{dt} = \frac{1}{1+t^2} }$$ $$\displaystyle{ \frac{d[\arccot(t)]}{dt} = -\frac{1}{1+t^2} }$$ $$\displaystyle{ \frac{d[\arcsec(t)]}{dt} = \frac{1}{\abs{t}\sqrt{t^2 -1}} }$$ $$\displaystyle{ \frac{d[\arccsc(t)]}{dt} = -\frac{1}{\abs{t}\sqrt{t^2 -1}} }$$

Trig Integrals

 $$\int{\sin(x)~dx} = -\cos(x)+C$$ $$\int{\cos(x)~dx} = \sin(x)+C$$ $$\int{\tan(x)~dx} = -\ln\abs{\cos(x)}+C$$ $$\int{\cot(x)~dx} = \ln\abs{\sin(x)}+C$$ $$\int{\sec(x)~dx} =$$ $$\ln\abs{\sec(x)+\tan(x)}+C$$ $$\int{\csc(x)~dx} =$$ $$-\ln\abs{\csc(x)+\cot(x)}+C$$

### Topics Listed Alphabetically

Single Variable Calculus

Multi-Variable Calculus

Differential Equations

Precalculus

Engineering

Circuits

Semiconductors

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 First and Second Derivatives Asymptotes Putting It All Together Practice

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