Calculus 3.
1 The Chain Rule Notes
Write your questions
and thoughts here!
Composite Functions:
𝑓 𝑔 𝑥
sin 𝑥 √ln 𝑥 cos sin 5𝑥
The Chain Rule (derivative of a composite function)
𝑑
𝑓 𝑔 𝑥 𝑓 𝑔 𝑥 ⋅𝑔 𝑥
𝑑𝑥
Find the derivative
1. 𝑓 𝑥 𝑥 5 2. 𝑔 𝑥 √4𝑥 3 3. ℎ 𝑥 sin 5𝑥
4. 𝑦 ln 𝑥 5. 𝑦 ln 𝑥
6. 𝑓 𝑡
Write your questions 7. If 𝑔 𝑥 2𝑥√1 𝑥 find 𝑔 3 .
and thoughts here!
8. Given the following table of values, find 𝑓′ 4 for each function.
𝑥 𝑔 𝑥 𝑔 𝑥 ℎ 𝑥 ℎ 𝑥
3 1 7 2 3
4 3 2 9 5
𝒇 𝒙 𝑔 𝑥 𝒇 𝒙 ℎ 𝑥 𝒇 𝒙 ℎ 𝑔 𝑥
3.1 The Chain Rule
Calculus
Practice
Find the derivative of each function.
1. 𝑔 𝑥 3𝑥 1 2. 𝑦 sin 2𝑥 3. ℎ 𝑟 √5𝑟 2𝑟 1
4. 𝑦 4 cos 𝑥 5. ℎ 𝑥 ln 5 6. 𝑔 𝑥 ln 2𝑥
7. 𝑓 𝑥 tan 2𝑥 8. 𝑦 cos 𝑥 9. 𝑦
10. 𝑓 𝑥 3√ 11. 𝑦 sin 4𝑥 12. 𝑦 𝑒√
13. 𝑔 𝑥 𝑒 14. ℎ 𝑥 sin ln 𝑥
Find 𝒇 𝟓 given the following.
𝑥 𝑔 𝑥 𝑔 𝑥 ℎ 𝑥 ℎ 𝑥
5 9 6 5 4
9 2 3 4 1
15. 𝑓 𝑥 ℎ 𝑔 𝑥 16. 𝑓 𝑥 ℎ 𝑥
17. 𝑓 𝑥 𝑔 𝑥 18. 𝑓 𝑥 2𝑔 𝑥 ℎ 𝑥
19. 𝑓 𝑥 20. 𝑓 𝑥 𝑔 ℎ 𝑥
Find the slope of the tangent line at the given 𝒙-value. Show work.
21. ℎ 𝑥 at 𝑥 2. 22. 𝑔 𝑥 cos tan 𝑥 at 𝑥 𝜋. 23. 𝑓 𝑥 1 𝑥 1 at 𝑥 2.
Find the equation of the tangent line at the given 𝒙-value.
24. 𝑓 𝑥 √𝑥 9 at 𝑥 5. 25. 𝑔 𝑥 𝑒 at 𝑥 1. 26. 𝑦 sin 3𝑥 at 𝑥 .
3.1 The Chain Rule Test Prep
27. The graph of the function 𝑓 is shown at the right. y
The function ℎ is defined by ℎ 𝑥 𝑓 2𝑥 𝑥 . Find the
𝑓 𝑥
slope of the line tangent to the graph of ℎ at the point where
𝑥 1
x
28. Let 𝑓 𝑥 2𝑒 and 𝑔 𝑥 5𝑥 . At what value of 𝑥 do the graphs of 𝑓 and 𝑔 have parallel
tangents?
(A) 0.445 (B) 0.366 (C) 0.344 (D) 0.251 (E) 0.165
29. Let 𝑓 be the function given by 𝑓 𝑥 5𝑒 . For what positive value of 𝑎 is the slope of the line
tangent to the graph of 𝑓 at 𝑎, 𝑓 𝑎 equal to 6?
(A) 0.142 (B) 0.344 (C) 0.393 (D) 0.595 (E) 0.714
30. Let 𝑓 𝑥 √2𝑥. If the rate of change of 𝑓 at 𝑥 𝑐 is four times its rate of change at 𝑥 1,
then 𝑐
(A) (B) (C) (D) 1 (E) 32
√ √
31. Let 𝑓 𝑥 𝑥∙𝑔 ℎ 𝑥 where 𝑔 4 2, 𝑔 4 3, ℎ 3 4, and ℎ 3 2. Find 𝑓′ 3 .
(A) 18 (B) 16 (C) 7 (D) 7 (E) 11
