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# Differentiation

David Jerison, Gigliola Staffilani, Jen French, Stephen Wang, and Jennifer French

How does the final velocity on a zip line change when the starting point is raised or lowered by a matter of centimeters? What is the accuracy of a GPS position measurement? How fast should an airplane travel to minimize fuel consumption? The answers to all of these questions involve the derivative.

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How does the final velocity on a zip line change when the starting point is raised or lowered by a matter of centimeters? What is the accuracy of a GPS position measurement? How fast should an airplane travel to minimize fuel consumption? The answers to all of these questions involve the derivative.

But what is the derivative? You will learn its mathematical notation, physical meaning, geometric interpretation, and be able to move fluently between these representations of the derivative. You will discover how to differentiate any function you can think up, and develop a powerful intuition to be able to sketch the graph of many functions. You will make linear and quadratic approximations of functions to simplify computations and gain intuition for system behavior. You will learn to maximize and minimize functions to optimize properties like cost, efficiency, energy, and power.

The three modules in this series are being offered as an XSeries on edX. Please visit Single Variable Calculus XSeries Program Page to learn more and to enroll in the modules.

Learn more about our High School and AP* Exam Preparation Courses

This course was funded in part by the Wertheimer Fund.

*Advanced Placement and AP are registered trademarks of the College Board, which was not involved in the production of, and does not endorse, these offerings.

What you'll learn

• How to evaluate limits graphically and numerically
• The physical meaning, and geometric interpretation of the derivative
• To calculate the derivative of any function
• To sketch many functions by hand
• To make linear and quadratic approximations of functions
• To apply derivatives to maximize and minimize functions and find related rates

## Good to know

Know what's good
, what to watch for
, and possible dealbreakers
Suitable for students in high school or preparing for the AP* Exam
Emphasizes the physical meaning and geometric interpretation of the derivative
Provides a strong foundation for further studies in mathematics

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## Reviews summary

### Excellent calculus course

Calculus 1A: Differentiation has received 5 reviews, all of which are 5-star reviews. Students are highly positive, reporting that the course materials are "excellent" and "very detailed and comprehensive". Overall, Calculus 1A: Differentiation appears to be a high-quality course that students find valuable.
Highly rated by students.
"An excellent course from MIT. Can't describe it but you have to take it to experience how good the course materials are."
Comprehensive course materials.
"Very detailed and comprehensive to the point where it's not possible not to understand concepts or not to be able to solve challenging problems."

## Career center

Learners who complete Calculus 1A: Differentiation will develop knowledge and skills that may be useful to these careers:
Physicist
Physicists study the fundamental laws of nature and the universe. They work on a wide range of projects, from developing new theories to designing new technologies. Calculus 1A provides Physicists with a solid foundation in calculus, which is essential for understanding the mathematical models used to describe physical phenomena. The course also helps students develop problem-solving skills and critical thinking abilities, which are essential for success in this field. Taking this course is highly recommended for anyone interested in a career in physics.
Aerospace Engineer
Aerospace Engineers incite the imagination with space travel, but they're also responsible for a lot more. They may be designing airplanes, satellites, or rockets; they might be managing construction or testing new products. The course Calculus 1A offers a foundation in calculus that is vital for Aerospace Engineers, providing them with the mathematical tools to calculate forces, predict trajectories and optimize designs. The skills and knowledge learned in this course can be applied to modeling the flight paths of aircraft, analyzing stress and strain on spacecraft structures, and optimizing rocket engine performance. For those aspiring to enter this field, taking this course will provide a strong foundation for success.
Mechanical Engineer
Mechanical Engineers design, develop, test, and supervise the installation of machines and mechanical systems. They work on a wide range of projects, from cars to airplanes to medical devices. Calculus 1A provides Mechanical Engineers with a solid foundation in calculus, which is essential for understanding the forces and stresses that act on mechanical systems. The course also helps students develop problem-solving skills and critical thinking abilities, which are essential for success in this field. Taking this course is highly recommended for anyone interested in a career in mechanical engineering.
Chemical Engineer
Chemical Engineers work on the design and maintenance of chemical plants. They create the processes that transform raw materials into useful products, such as fuel, pharmaceuticals, and plastics. Calculus 1A provides Chemical Engineers with a strong foundation in calculus, which is essential for understanding the physical and chemical processes involved in chemical engineering. Students will use the concepts learned in this course to solve complex problems related to fluid flow, heat transfer, and chemical reactions. Taking this course is highly recommended for anyone interested in a career in chemical engineering.
Geologist
Geologists study the Earth's physical structure and history. They work on a wide range of projects, from oil exploration to environmental remediation. Calculus 1A provides Geologists with a solid foundation in calculus, which is essential for understanding the mathematical models used to analyze geological data. The course also helps students develop problem-solving skills and critical thinking abilities, which are essential for success in this field. Taking this course is highly recommended for anyone interested in a career in geology.
Civil Engineer
Civil Engineers are responsible for designing and constructing infrastructure, including roads, bridges, buildings, and water systems. They must be able to understand and apply the principles of mathematics and physics to ensure that their structures are safe and efficient. Calculus 1A provides Civil Engineers with a solid foundation in calculus, which is essential for understanding the forces and stresses that act on structures. The course also helps students develop problem-solving skills and critical thinking abilities, which are essential for success in this field. Taking this course is highly recommended for anyone interested in a career in civil engineering.
Electrical Engineer
Electrical Engineers design, develop, test, and supervise the installation of electrical systems and equipment. They work on a wide range of projects, from power plants to telecommunications networks. Calculus 1A provides Electrical Engineers with a solid foundation in calculus, which is essential for understanding the electrical circuits and systems they design. The course also helps students develop problem-solving skills and critical thinking abilities, which are essential for success in this field. Taking this course is highly recommended for anyone interested in a career in electrical engineering.
Operations Research Analyst
Operations Research Analysts use mathematical and statistical techniques to solve problems in business and industry. They work on a wide range of projects, from improving supply chain efficiency to reducing manufacturing costs. Calculus 1A provides Operations Research Analysts with a solid foundation in calculus, which is essential for understanding the mathematical models used to solve business problems. The course also helps students develop problem-solving skills and critical thinking abilities, which are essential for success in this field. Taking this course is highly recommended for anyone interested in a career in operations research.
Statistician
Statisticians collect, analyze, interpret, and present data. They work on a wide range of projects, from designing clinical trials to conducting market research. Calculus 1A provides Statisticians with a solid foundation in calculus, which is essential for understanding the statistical methods used to analyze data. The course also helps students develop problem-solving skills and critical thinking abilities, which are essential for success in this field. Taking this course is highly recommended for anyone interested in a career in statistics.
Data Scientist
Data Scientists are responsible for collecting, analyzing, and interpreting data to help businesses make better decisions. They must have a strong foundation in mathematics and statistics, as well as the ability to program computers. Calculus 1A provides Data Scientists with a solid foundation in calculus, which is essential for understanding the statistical methods used to analyze data. The course also helps students develop problem-solving skills and critical thinking abilities, which are essential for success in this field. Taking this course is highly recommended for anyone interested in a career in data science.
Financial Analyst
Financial Analysts use mathematical and statistical techniques to analyze financial data and make investment recommendations. They must have a strong foundation in mathematics and economics, as well as the ability to communicate their findings clearly and concisely. Calculus 1A provides Financial Analysts with a solid foundation in calculus, which is essential for understanding the financial models used to analyze investments. The course also helps students develop problem-solving skills and critical thinking abilities, which are essential for success in this field. Taking this course is highly recommended for anyone interested in a career in financial analysis.
Software Engineer
Software Engineers design, develop, test, and maintain software systems. They work on a wide range of projects, from developing operating systems to creating mobile apps. Calculus 1A provides Software Engineers with a solid foundation in calculus, which is essential for understanding the mathematical models used to design and analyze software systems. The course also helps students develop problem-solving skills and critical thinking abilities, which are essential for success in this field. Taking this course is highly recommended for anyone interested in a career in software engineering.
Teacher
Teachers lead classroom instruction of students at various grade levels, and work to help students reach their full potential. They create lesson plans, assess student performance, and provide students with feedback. Calculus 1A can be useful for Teachers of mathematics by strengthening your understanding of calculus and its applications, which can help you create engaging and effective lesson plans and provide your students with a deeper understanding of the subject.

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