Differentiation from Tables, Graphs
3.2 and Symbolically Homework
Let 𝑓𝑓 𝑥𝑥 , 𝑓𝑓 ′ 𝑥𝑥 , 𝑔𝑔 𝑥𝑥 , 𝑔𝑔′ 𝑥𝑥 be differentiable functions whose values are given in the table below. Find
the symbolic differentiation. Then use the values from the table to find the derivative at the designated
point.
𝒙𝒙 −1 0 1 2
𝒇𝒇(𝒙𝒙) −1 3 −2 0
𝒇𝒇′ (𝒙𝒙) −3 4 1 5
𝒈𝒈(𝒙𝒙) 2 1 5 −2
𝒈𝒈′ (𝒙𝒙) 7 −3 0 −1
𝑑𝑑 𝑑𝑑 2
1. 𝑑𝑑𝑑𝑑
𝑓𝑓 + 𝑔𝑔 Evaluate when 𝑥𝑥 = 1 2. 𝑑𝑑𝑑𝑑
𝑓𝑓 − 𝑔𝑔 Evaluate when 𝑥𝑥 = 0
𝑑𝑑 1 𝑑𝑑
3. Evaluate when 𝑥𝑥 = 2 4. 𝑑𝑑𝑑𝑑
𝑓𝑓 + 𝑔𝑔 Evaluate when 𝑥𝑥 = 1
𝑑𝑑𝑑𝑑 𝑓𝑓−𝑔𝑔
𝑑𝑑 𝑑𝑑
5. 𝑑𝑑𝑑𝑑
𝑓𝑓 � 𝑔𝑔 Evaluate when 𝑥𝑥 = 0 6. 𝑑𝑑𝑑𝑑
𝑓𝑓 2 Evaluate when 𝑥𝑥 = 2
𝑑𝑑 1
−1 𝑑𝑑
7. 𝑑𝑑𝑑𝑑
𝑓𝑓𝑓𝑓 Evaluate when 𝑥𝑥 = −1 8. 𝑓𝑓𝑓𝑓 2 Evaluate when 𝑥𝑥 = 0
𝑑𝑑𝑑𝑑
𝑑𝑑 𝑔𝑔 𝑑𝑑
9. Evaluate when 𝑥𝑥 = 1 10. 𝑑𝑑𝑑𝑑
(𝑓𝑓 𝑔𝑔 𝑥𝑥 Evaluate when 𝑥𝑥 = −1
𝑑𝑑𝑑𝑑 𝑓𝑓
𝑑𝑑 𝑑𝑑
11. 𝑑𝑑𝑑𝑑
𝑓𝑓 − 𝑔𝑔 Evaluate when 𝑥𝑥 = −1 12. 𝑑𝑑𝑑𝑑
𝑔𝑔𝑔𝑔 Evaluate when 𝑥𝑥 = 1
𝑑𝑑 2 𝑑𝑑 3
13. 𝑑𝑑𝑑𝑑
𝑓𝑓𝑓𝑓 Evaluate when 𝑥𝑥 = 0 14. 𝑑𝑑𝑑𝑑
𝑓𝑓 � 𝑔𝑔 Evaluate when 𝑥𝑥 = −1
𝑑𝑑 𝑓𝑓 𝑑𝑑 1
15. 𝑑𝑑𝑑𝑑 3 Evaluate when 𝑥𝑥 = 0 16. 𝑑𝑑𝑑𝑑 𝑓𝑓 2
Evaluate when 𝑥𝑥 = 0
𝑔𝑔
Problems 17 – 20, use the table of values to calculate the derivative of the function at the given
point.
𝒙𝒙 𝒇𝒇 𝒙𝒙 𝒇𝒇′ 𝒙𝒙 𝒈𝒈(𝒙𝒙) 𝒈𝒈′ (𝒙𝒙)
1 3 4 9 −2
3 0 −1 2 3
6 −1 1/2 3 4
17. 𝐻𝐻 𝑥𝑥 = 𝑓𝑓 𝑔𝑔 𝑥𝑥 . Find 𝐻𝐻 ′ (6) 18. 𝐾𝐾 𝑥𝑥 = 𝑔𝑔 𝑥𝑥 . Find 𝐾𝐾 ′ (9)
19. 𝐽𝐽(𝑥𝑥) = 𝑒𝑒 𝑓𝑓 𝑥𝑥
. Find 𝐽𝐽′ (3) 20. 𝑅𝑅 𝑥𝑥 = ln 𝑥𝑥 2 � 𝑔𝑔 𝑥𝑥 . Find 𝑅𝑅 ′ (1)
Problems 21– 22, let 𝑯𝑯 𝒙𝒙 = 𝒇𝒇 𝒈𝒈 𝒙𝒙 , where the graphs
of 𝒇𝒇 and 𝒈𝒈 are shown at right. Estimate each derivative.
Justify your analysis algebraically.
21. 𝑔𝑔′ 𝑓𝑓 6 22. 𝐻𝐻 ′ (6)
