In this video lecture, Prof. Gilbert Strang discusses Chains f(g(x)) and the Chain Rule.
A chain of functions starts with y = g(x). Then it finds z = f(y). So z = f(g(x)). Very many functions are built this way, g inside f . So we need their slopes.The Chain Rule says: Multiply the Slopes of f and g.Find dy/dx for g(x). Then find dz/dy for f(y).Since dz/dy is found in terms of y, substitute g(x) in place of y.The way to remember the slope of the chain is dz/dx = dz/dy times dy/dx.Remove y to get a function of x. The slope of z = sin (3x) is 3 cos (3x).

1. Faculty Introduction
2. Big Picture of Calculus
3. Big Picture: Derivatives
4. Max and Min and Second Derivative
5. The Exponential Function
6. Big Picture: Integrals
7. Derivative of sin x and cos x
8. Product Rule and Quotient Rule
9. Chains f(g(x)) and the Chain Rule
10. Limits and Continuous Functions
11. Inverse Functions f ^-1 (y) and the Logarithm x = ln y
12. Derivatives of ln y and sin ^-1 (y)
13. Growth Rate and Log Graphs
14. Linear Approximation/Newton's Method
15. Power Series/Euler's Great Formula
16. Differential Equations of Motion
17. Differential Equations of Growth
18. Six Functions, Six Rules, and Six Theorems

Highlights of Calculus is a series of short videos that introduces the basics of calculus—how it works and why it is important. The intended audience is high school students, college students, or anyone who might need help understanding the subject.

The series is divided into three sections:
Introduction
- Why Professor Strang created these videos
- How to use the materials

Highlights of Calculus
- Five videos reviewing the key topics and ideas of calculus
- Applications to real-life situations and problems
- Additional summary slides and practice problems

Derivatives
- Twelve videos focused on differential calculus
- More applications to real-life situations and problems
- Additional summary slides and practice problems

Acknowledgements
Special thanks to Professor J.C. Nave for his help and advice on the development and recording of this program.The video editing was funded by the Lord Foundation of Massachusetts.