# Lecture 11: Differentials and the chain rule

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Lesson Summary:

In this lesson on differentials and the chain rule, we learn about a new tool called the total differential, which includes all the contributions that can cause the value of a function to change. We also explore how to think of differentials as placeholders to put values and obtain tangent approximation formulas. The chain rule is introduced as a way to find the rate of change of a function on a new variable in terms of its derivatives and the dependence between the variables. Examples are given to demonstrate how to apply these concepts in calculations.

Lesson Description:

Learn about differentials and the chain rule in many variables.

Denis Auroux. 18.02 Multivariable Calculus, Fall 2007. (Massachusetts Institute of Technology: MIT OpenCourseWare). http://ocw.mit.edu (Accessed March 15, 2009). License: Creative Commons Attribution-Noncommercial-Share Alike.

• What are differentials and how do you compute them?
• What is implicit differentiation?
• What is a total differential and how do you find it?
• How is df different than delta f?
• Why are differentials useful?
• What is the chain rule for differentials?
• How can you use the chain rule to find dw/dt when w = x^y + z, x = t, y = e^t, and z = sint?
• What is the product rule and quotient rule using differentials?
• How do you use the chain rule with four variables?
• How do you find df/du and df/dv if u and v depend on x and y?
• How does the chain rule work with parametric functions?
• How does the chain rule work with polar coordinates?
• What is the gradient vector and how do you find it?
• #### Staff Review

• Currently 4.0/5 Stars.
Differentials are the topic in this lecture, and they are explained both in theory and in practical problems. Differentials are different than delta of the variable. This is an important lecture in understanding the theory behind topics in this course. The chain rule is discussed in depth with actual problems for functions of multiple variables.