Back To Course

Math 104: Calculus14 chapters | 115 lessons | 11 flashcard sets

Watch short & fun videos
**
Start Your Free Trial Today
**

Start Your Free Trial To Continue Watching

As a member, you'll also get unlimited access to over 70,000 lessons in math, English, science, history, and more. Plus, get practice tests, quizzes, and personalized coaching to help you succeed.

Free 5-day trial
Your next lesson will play in
10 seconds

Lesson Transcript

Instructor:
*Eric Garneau*

Differentiating functions doesn't have to stop with the first or even second derivative. Learn what a mathematical jerk is as you calculate derivatives of any order in this lesson.

I love riding on roller coasters. One of my favorite roller coasters of all time is Space Mountain. On this roller coaster you're kept in the dark the entire time. So picture it: you're going along in the dark, and suddenly you're jerked to one side, then jerked to the other as you careen around this Space Mountain. It's fantastic! But as I'm being jerked around, I always think about one thing: Why do we use the term 'jerk'? What in the world does that mean?

Well, it goes back to **derivatives** and the rate of change. If you have some function like *y*=*f(t)*, the position, *y*, is a function of time, *t*. Say this is my height on the roller coaster. Then I can look at *y`*(*dy/dt*), which is the rate of change, *d/dt*, of my position, *y*. So this rate of change is my **velocity**, it's how fast my height is changing as a function of time. I could take the derivative of that, *y``*, or ((*d*^2)*y*)/*dt*^2, as the derivative of the rate of change of position, so it's the derivative of the velocity. And the derivative of the velocity is the **acceleration**. Well, the acceleration can also change, so I can write *y```*, and that's the rate of change of the acceleration. And how fast my acceleration changes is known as the **jerk**.

So you know how on a roller coaster you're completely stopped at first? There's no acceleration, no velocity, no nothing. Then all of a sudden you jerk, or lurch, forward. That's a change in your acceleration; that's *d/dt* of your acceleration. So the key here is that the derivative is just a rate of change. But in the real world, nothing is static. Everything is dynamic; everything changes. Static means stationary and unchanging, and dynamic means changing. You measure this change using derivatives.

Let's do an example. Let's say we have position, *f(t)*, as a function of time, *t*, and it equals sin(*t*) + *t*^3. I know that the velocity is the derivative of the position, so *f`(t)* is *d/dt* sin(*t*) + *t*^3. That's my position. I can find this derivative by first dividing and conquering, so *d/dt* sin(*t*) + *d/dt*(*t*^3). Well using my derivative rules here, *d/dt* sin(*t*)=cos(*t*), and *d/dt*(*t*^3) is 3*t*^2, so my velocity is cos(*t*) + 3*t*^2. My acceleration is the derivative of the velocity - it's how fast my velocity is changing - and that's *f``*, or *d/dt* *f`(t)*. I can calculate this by finding the derivative, *d/dt*, of my velocity, which is cos(*t*) + 3*t*^2. Again I can divide and conquer to get *d/dt* cos(*t*) + 3(*d/dt*)*t*^2. Then using my derivative rules, I find that the acceleration is -sin(*t*) + 3(2*t*), or -sin(*t*) + 6*t*. Now that I know the acceleration, I can find the jerk, which is just the derivative or the rate of change of the acceleration. This is *f```*, or *d/dt* *f``(t)*, so that's *d/dt* of the acceleration, the rate of change of the acceleration. I can calculate that by finding *d/dt* of the acceleration, which is -sin(*t*) + 6*t*. Divide and conquer; that equals -(*d/dt*)sin(*t*) + 6(*d/dt*)*t*. Again using my rules I know that this equals -cos(*t*) + 6. So in this case, where my position was originally sin(*t*) + *t*^3, the jerk as a function of time - that's how fast my acceleration is changing as a function of time - is equal to -cos(*t*) + 6.

We can use these same principles to find any higher-order derivative. So, for example, we can find the fourth-order derivative of *f(x)* = *x*^(-1) + cos(4*x*). This fourth-order derivative is *f````*. Mathematicians kind of get lazy after the first three, so we write *f*^4. Let's find the fourth-order derivative of this function, *f(x)* = *x*^(-1) + cos(4*x*).

*f`(x)* is the derivative of this function, so I'm going to divide and conquer and find that *f`(x)*= -*x*^(-2) - 4(sin(4*x*)), because here I have to use the chain rule. Once I have the first derivative, *f`(x)*, I can find the second derivative, *f``(x)*, and that's the derivative, *d/dx* of my first derivative, or *d/dx*(-*x*^(-2) - 4(sin(4*x*)). I can divide and conquer and use my differentiation rules to find that this second-order derivative is 2*x*^(-3) - 16cos(4*x*). So now we can keep going. *f```(x)* is the derivative of *f``(x)*, so that's *d/dx*(2*x*^(-3) - 16cos(4*x*)). Calculating this out using our divide and conquer/differentiation rules, we find that this derivative, this *f```(x)*, equals -6*x*^(-4) + 64sin(4*x*). I've got the third-order derivative, but I still need the fourth-order derivative, so let's take one more differentiation. *f*^4(*x*) is the derivative *d/dx*(*f```(x)*), or -6*x*^(-4) + 64sin(4*x*). I can divide this and conquer, so take the derivative of -6*x*^(-4), and I can add that to the derivative of 64sin(4*x*). I have to use the chain rule, and I find that my fourth-order derivative is 64*x*^(-5) + 256cos(4*x*).

Let's review. The **jerk** is kind of like what you have on Space Mountain; it's the derivative of the acceleration or the rate of change of the acceleration, how your acceleration is changing as a function of time. If you graph this, it's the slope of the tangent of your acceleration as a function of time. It is also the third derivative, *f```(t)*, of your position, *f(t)*. In calculating the jerk, we also learn some important things about higher-order derivatives. To find higher-order derivatives - the second derivative, the third derivative, the fourth derivative, etc. - just keep differentiating. You differentiate *f* to get *f`*, you differentiate *f`* to get *f``*, you differentiate that to get *f```*, and so on and so forth. You can calculate as high as you want, even say up to the 47th-order derivative. Just keep differentiating.

To unlock this lesson you must be a Study.com Member.

Create
your account

Already a member? Log In

BackDid you know… We have over 95 college courses that prepare you to earn credit by exam that is accepted by over 2,000 colleges and universities. You can test out of the first two years of college and save thousands off your degree. Anyone can earn credit-by-exam regardless of age or education level.

To learn more, visit our Earning Credit Page

Not sure what college you want to attend yet? Study.com has thousands of articles about every imaginable degree, area of study and career path that can help you find the school that's right for you.

You are viewing lesson
Lesson
10 in chapter 7 of the course:

Back To Course

Math 104: Calculus14 chapters | 115 lessons | 11 flashcard sets

- Go to Continuity

- Go to Limits

- Using Limits to Calculate the Derivative 8:11
- The Linear Properties of a Derivative 8:31
- Calculating Derivatives of Trigonometric Functions 7:20
- Calculating Derivatives of Polynomial Equations 10:25
- Calculating Derivatives of Exponential Equations 8:56
- Using the Chain Rule to Differentiate Complex Functions 9:40
- Differentiating Factored Polynomials: Product Rule and Expansion 6:44
- When to Use the Quotient Rule for Differentiation 7:54
- Understanding Higher Order Derivatives Using Graphs 7:25
- Calculating Higher Order Derivatives 9:24
- How to Calculate Derivatives of Inverse Trigonometric Functions 7:48
- Applying the Rules of Differentiation to Calculate Derivatives 11:09
- Optimization Problems in Calculus: Examples & Explanation 10:45
- Go to Calculating Derivatives and Derivative Rules

- Psychology 316: Advanced Social Psychology
- Hiring & Developing Employees
- Accounting 305: Auditing & Assurance Services
- MTEL Physical Education (22): Study Guide & Test Prep
- Praxis Art - Content Knowledge (5134): Practice & Study Guide
- Victimization at School & the Workplace
- Types of Property Crimes
- The Victim Movement
- Consequences of Crime Victimization
- The Nature of Violence
- Study.com CLEP Scholarship for Military Members
- Study.com Scholarship for Texas Students & Prospective Teachers
- Study.com Scholarship for Florida Students & Prospective Teachers
- What are TExMaT Exams?
- What is the Florida Teacher Certification Examination (FTCE)?
- Study.com TExES Scholarship: Application Form & Information
- Study.com FTCE Scholarship: Application Form & Information

- Children Living in Poverty: Facts, Effects & Statistics
- Reading & Interpreting Bar Graphs
- Specific Intent Crimes: Definition & Examples
- What is DNA Fingerprinting? - Process & Uses
- How to Evaluate International Markets
- Causation of War: Individual, State & System
- The Mesolithic Age in India
- Rural Settlement Pattern Types
- Quiz & Worksheet - Conspiracy in Law
- Quiz & Worksheet - Pantheistic Beliefs & Religions
- Quiz & Worksheet - Spanish Adjectives Starting With S
- Quiz & Worksheet - Benefits to Reflective Practices at Work
- Quiz & Worksheet - Methods of DNA Sequencing
- International Law & Global Issues Flashcards
- Foreign Policy, Defense Policy & Government Flashcards

- Geography: Middle School
- NES Business Education (309): Practice & Study Guide
- NES Essential Academic Skills Mathematics Subtest 3 (003): Practice & Study Guide
- Pathophysiology Syllabus Resource & Lesson Plans
- Ohio Assessments for Educators - Marketing: Practice & Study Guide
- Overview of Medical Terminology
- The Neolithic Age
- Quiz & Worksheet - Court of Last Resort
- Quiz & Worksheet - Degradation Ceremony
- Quiz & Worksheet - Chorionic Villi
- Quiz & Worksheet - Probation Rules & Types
- Quiz & Worksheet - Road Rage: Causes & Effects

- Crisis Situations: Types & Examples
- Ascaris Worms: Life Cycle & Pictures
- How to Pass the CCRN Exam
- How to Pass the PE Exam
- EPT Test Dates
- Engineering Competitions for High School
- Community Lesson Plan
- Free English Language Courses
- TExES PPR Test Dates
- What To Do If Your School Doesn't Accept Study.com Credit
- How to Pass the FE Exam
- How to Pass Algebra 1

Browse by subject