1. The Chain Rule

2. Objective To use the chain rule for differentiation.
ES: Explicitly assess information and draw conclusions

3. The Chain Rule
The Chain Rule is a technique for differentiating composite functions.
Composite functions are made up of layers of functions inside of functions.

4. The Chain Rule
The outside

5. The Chain Rule
Peel it away

6. The Chain Rule
There’s still
more to peel.

7. The Chain Rule Some functions have many layers that must
be peeled away in order
to find their derivatives.

8. The Chain Rule

9. The Chain Rule
Inside function
Outside function

10. The Chain Rule

11. The Chain Rule 1. Identify inner and outer functions.
2. Derive outer function, leaving the inner function alone.
3. Derive the inner function.

12. The Chain Rule (Example A)

13. The Chain Rule
Inside function
Key Point: If the inside function contains something other than plain old “x,” you must use the Chain Rule to find the derivative.

14. The Chain Rule
Outside function

15. The Chain Rule
Inside function

16. The Chain Rule
Outside function

17. The Chain Rule

18. The Chain Rule The derivative of the outside The derivative
of the inside (blop)

19. The Chain Rule

20. Possible mistakes ahead!
The Chain Rule
CAUTION:
Possible mistakes ahead!
What mistake did I make?
I changed the inside function and did not multiple
by the derivative of the inside function.

21. The Chain Rule (Example B)

22. The Chain Rule (Example C)

23. The Chain Rule (Example D)

24. The Chain Rule (Example E)

25. The Chain Rule (Example F)

26. The Chain Rule (Example G)

27. The Chain Rule (Example H)

28. The Chain Rule (Example I)

29. Conclusion
Remember: When a function is inside another function, use the Chain Rule to find the derivative.
First, differentiate the outside function, leaving the inside function alone.
Last, differentiate the inside function.