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Calculus

Single Variable Part 2 - Differentiation

Robert Ghrist

Calculus is one of the grandest achievements of human thought, explaining everything from planetary orbits to the optimal size of a city to the periodicity of a heartbeat. This brisk course covers the core ideas of single-variable Calculus with emphases on conceptual understanding and applications. The course is ideal for students beginning in the engineering, physical, and social sciences. Distinguishing features of the course include: 1) the introduction and use of Taylor series and approximations from the beginning; 2) a novel synthesis of discrete and continuous forms of Calculus; 3) an emphasis on the conceptual over the computational; and 4) a clear, dynamic, unified approach.

In this second part--part two of five--we cover derivatives, differentiation rules, linearization, higher derivatives, optimization, differentials, and differentiation operators.

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What's inside

Syllabus

A New Look at Differentiation

Think derivatives mean "slopes"? Not anymore... In this module, we will reconsider what a derivative is and means in terms of the asymptotic (or big-O) notation from the previous chapter. This will give us a new language for describing and understanding rates of change and the rules that govern them.

Putting Derivatives to Work

Why exactly are derivatives so central to calculus? In part, it is because they are so ubiquitously useful! In this module, we will recall a few core applications of derivatives. In so doing, we'll see exactly how having an understanding of the asymptotics assists in building applications of the derivative.

Differentials and Operators

There is much more to derivatives than simply their computation and applications. So much of how they arise is calculus is in the mysterious guise of *differentials*. These arise from implicit differentiation, which in turn reveals a deeper level of understanding of what differentiation means.

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Provides a clear understanding of calculus concepts with emphasis on conceptual understanding and applications

Covers core ideas of single-variable Calculus, making it suitable for students in engineering, physical, and social sciences

Incorporates Taylor series and approximations from the beginning, providing a deeper understanding of calculus

Introduces a novel synthesis of discrete and continuous forms of Calculus, enhancing conceptual understanding

Emphasizes conceptual understanding over computational aspects, facilitating a deeper grasp of the subject

Facilitates a clear and dynamic approach to learning Calculus, making it accessible and engaging

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Reviews summary

Highly rated calculus differentiation course

According to students, Calculus: Single Variable Part 2 - Differentiation is a well-received course. The course features engaging assignments, lectures from knowledgeable instructors, and clear explanations with helpful visuals. Students report that the course is challenging but well organized, making it a suitable choice for those with a strong foundation in calculus.

Activities

Be better prepared before your course. Deepen your understanding during and after it. Supplement your coursework and achieve mastery of the topics covered in Calculus: Single Variable Part 2 - Differentiation with these activities:

Review prerequisite math

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Calculus relies on a strong foundation in mathematics. Revisiting prerequisite math will ensure you have a strong foundation in place before beginning this course.

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Review basic algebra concepts, such as linear equations, quadratic equations, and polynomials.
Practice solving trigonometric equations and identities.
Review the concept of limits and how they are used to define derivatives.

Review key concepts from prerequisite courses

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Calculus builds on concepts from prerequisite courses. Reviewing these concepts will strengthen your foundation and make learning calculus easier.

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Review your notes or textbooks from prerequisite courses.
Focus on understanding the core concepts and their relevance to calculus.
Practice solving problems related to these concepts.

Form a study group with classmates

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Collaboration and discussion can enhance understanding. A study group provides a platform to share knowledge, ask questions, and learn from peers.

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Identify classmates who are also taking the course.
Schedule regular study sessions to discuss course material and work on problems together.
Take turns explaining concepts and solving problems.

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Follow online tutorials on differentiation rules

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Differentiation rules are essential for solving derivatives. Tutorials provide a structured approach to learning and practicing these rules.

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Find online tutorials that cover the chain rule, product rule, and quotient rule.
Follow the tutorials and take notes on the steps involved in applying each rule.
Practice applying the rules to solve differentiation problems.

Practice differentiating functions

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Practice is key to mastering differentiation. This activity provides an opportunity to apply your understanding and improve your skills.

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Find a collection of differentiation problems.
Attempt to solve the problems on your own.
Check your answers and identify areas where you need improvement.

Create visual aids for understanding derivatives

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Visual aids can enhance understanding and retention. Creating your own visual aids will reinforce your learning and provide a valuable resource for review.

Browse courses on Differentiation

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Identify key concepts in differentiation that you find challenging.
Create diagrams, graphs, or other visual representations to illustrate these concepts.
Use your visual aids to explain differentiation to a peer or family member.

Apply differentiation to real-world problems

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Applying differentiation to real-world problems demonstrates its practical significance. This project will challenge you to use your understanding to solve meaningful problems.

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Identify a real-world problem that can be solved using differentiation.
Develop a mathematical model that describes the problem.
Use differentiation to find the solution to the problem.

Career center

Learners who complete Calculus: Single Variable Part 2 - Differentiation will develop knowledge and skills that may be useful to these careers:

Data Scientist

A Data Scientist uses mathematical and statistical techniques to extract knowledge from data. This course can help build a foundation for this career by providing a deep understanding of calculus, which is essential for data analysis and modeling. The course's emphasis on conceptual understanding and applications will also be beneficial for Data Scientists who need to apply calculus to real-world problems.

See salaries and explore the career path for Data Scientist

Software Engineer

Software Engineers design, develop, and maintain software applications. This course can help build a foundation for this career by providing a deep understanding of calculus, which is essential for developing efficient and reliable software. The course's emphasis on conceptual understanding and applications will also be beneficial for Software Engineers who need to apply calculus to real-world problems.

See salaries and explore the career path for Software Engineer

Financial Analyst

Financial Analysts use mathematical and statistical techniques to analyze financial data and make investment recommendations. This course can help build a foundation for this career by providing a deep understanding of calculus, which is essential for financial modeling and analysis. The course's emphasis on conceptual understanding and applications will also be beneficial for Financial Analysts who need to apply calculus to real-world problems.

See salaries and explore the career path for Financial Analyst

Operations Research Analyst

Operations Research Analysts use mathematical and statistical techniques to solve business problems. This course can help build a foundation for this career by providing a deep understanding of calculus, which is essential for developing and analyzing mathematical models. The course's emphasis on conceptual understanding and applications will also be beneficial for Operations Research Analysts who need to apply calculus to real-world problems.

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Actuary

Actuaries use mathematical and statistical techniques to assess risk and uncertainty. This course can help build a foundation for this career by providing a deep understanding of calculus, which is essential for developing and analyzing actuarial models. The course's emphasis on conceptual understanding and applications will also be beneficial for Actuaries who need to apply calculus to real-world problems.

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Statistician

Statisticians use mathematical and statistical techniques to collect, analyze, and interpret data. This course can help build a foundation for this career by providing a deep understanding of calculus, which is essential for developing and analyzing statistical models. The course's emphasis on conceptual understanding and applications will also be beneficial for Statisticians who need to apply calculus to real-world problems.

See salaries and explore the career path for Statistician

Economist

Economists use mathematical and statistical techniques to analyze economic data and make policy recommendations. This course can help build a foundation for this career by providing a deep understanding of calculus, which is essential for developing and analyzing economic models. The course's emphasis on conceptual understanding and applications will also be beneficial for Economists who need to apply calculus to real-world problems.

See salaries and explore the career path for Economist

Physicist

Physicists use mathematical and statistical techniques to study the laws of nature. This course can help build a foundation for this career by providing a deep understanding of calculus, which is essential for developing and analyzing physical models. The course's emphasis on conceptual understanding and applications will also be beneficial for Physicists who need to apply calculus to real-world problems.

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Engineer

Engineers use mathematical and statistical techniques to design and build structures and machines. This course can help build a foundation for this career by providing a deep understanding of calculus, which is essential for developing and analyzing engineering models. The course's emphasis on conceptual understanding and applications will also be beneficial for Engineers who need to apply calculus to real-world problems.

See salaries and explore the career path for Engineer

Mathematician

Mathematicians use mathematical and statistical techniques to study the nature of numbers, quantities, and space. This course can help build a foundation for this career by providing a deep understanding of calculus, which is a core area of mathematics. The course's emphasis on conceptual understanding and applications will also be beneficial for Mathematicians who need to apply calculus to real-world problems.

See salaries and explore the career path for Mathematician

Computer Scientist

Computer Scientists use mathematical and statistical techniques to design and develop computer systems. This course can help build a foundation for this career by providing a deep understanding of calculus, which is essential for developing and analyzing computer algorithms. The course's emphasis on conceptual understanding and applications will also be beneficial for Computer Scientists who need to apply calculus to real-world problems.

See salaries and explore the career path for Computer Scientist

Biostatistician

Biostatisticians use mathematical and statistical techniques to analyze biological data. This course can help build a foundation for this career by providing a deep understanding of calculus, which is essential for developing and analyzing biostatistical models. The course's emphasis on conceptual understanding and applications will also be beneficial for Biostatisticians who need to apply calculus to real-world problems.

See salaries and explore the career path for Biostatistician

Quantitative Analyst

Quantitative Analysts use mathematical and statistical techniques to analyze financial data and make investment recommendations. This course can help build a foundation for this career by providing a deep understanding of calculus, which is essential for developing and analyzing quantitative models. The course's emphasis on conceptual understanding and applications will also be beneficial for Quantitative Analysts who need to apply calculus to real-world problems.

See salaries and explore the career path for Quantitative Analyst

Market Researcher

Market Researchers use mathematical and statistical techniques to collect and analyze data about consumer behavior. This course can help build a foundation for this career by providing a deep understanding of calculus, which is essential for developing and analyzing market research models. The course's emphasis on conceptual understanding and applications will also be beneficial for Market Researchers who need to apply calculus to real-world problems.

See salaries and explore the career path for Market Researcher

Epidemiologist

Epidemiologists use mathematical and statistical techniques to study the causes and patterns of disease. This course can help build a foundation for this career by providing a deep understanding of calculus, which is essential for developing and analyzing epidemiological models. The course's emphasis on conceptual understanding and applications will also be beneficial for Epidemiologists who need to apply calculus to real-world problems.

See salaries and explore the career path for Epidemiologist

Reading list

We've selected 15 books that we think will supplement your learning. Use these to develop background knowledge, enrich your coursework, and gain a deeper understanding of the topics covered in Calculus: Single Variable Part 2 - Differentiation.

Calculus: Early Transcendentals

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Classic calculus textbook that provides a comprehensive overview of the subject. It is well-written and easy to follow, and it provides a good foundation for students who are new to calculus.

Calculus

Single Variable Part 2 - Differentiation

Here's a deal for you

What's inside

Syllabus

Good to know

Save this course

Reviews summary

Highly rated calculus differentiation course

Activities

Career center

Reading list

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