Lec 11: Differentials; chain rule | MIT 18.02 Multivariable Calculus, Fall 2007

TL;DR

This content explains the concept of differentials and how to use them in functions with several variables. It also introduces the chain rule and its applications in differentiating functions.

Transcript

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Key Insights

• 💱 Differentials represent changes in functions of multiple variables and cannot be assigned specific values like regular numbers.
• 📏 Differentiation rules, such as the product rule and quotient rule, can be derived using differentials and the chain rule.
• ❓ Differentiating functions of multiple variables requires using partial derivatives and considering how each variable depends on others.
• 💱 The total differential includes all the factors that can cause a function's value to change, while partial derivatives only account for changes in one variable at a time.
• 📏 The chain rule is a powerful tool in finding the derivative of composite functions, especially when the function depends on multiple variables that themselves depend on another parameter.
• ☠️ The rate of change of a function with respect to a parameter can be found by multiplying the partial derivatives with the corresponding rates of change of the variables that depend on the parameter.

Questions & Answers

Q: What is the difference between the total differential and partial derivatives?

The total differential includes all the contributions that can cause a function's value to change, while partial derivatives only consider changes in one variable while keeping the others constant. The total differential is denoted by "df" and can be expressed as a sum of the partial derivatives multiplied by the differentials of each variable.

Q: How are differentials used to estimate changes in functions?

Differentials can be used to approximate the change in a function when its variables change by a small amount. By expressing the function as a sum of the partial derivatives multiplied by the differentials of each variable, we can estimate the change in the function by multiplying the partial derivatives with the corresponding changes in variables.

Q: What is the chain rule and why is it important?

The chain rule is a method for finding the derivative of a composite function. It is important because it allows us to find the rate of change of a function with respect to a parameter, even when the function depends on multiple variables that themselves depend on that parameter. The chain rule helps us understand how changes in one variable affect the overall function.

Q: How are differentials and partial derivatives related to each other?

Differentials and partial derivatives are both used to analyze functions with multiple variables. Differentials provide a way to estimate changes in a function, while partial derivatives measure the rate of change of a function with respect to one variable while keeping others constant. Differentials are expressed as a sum of the partial derivatives multiplied by the differentials of each variable.

Summary & Key Takeaways

• The content explains the concept of differentials and their significance in studying functions of several variables.

• It discusses how to use differentials to estimate changes in functions, as well as their relation to partial derivatives.

• The content introduces the chain rule and its applications in finding the rate of change of a function with respect to a parameter.