Calculus 2.
7 Derivatives of 𝐜𝐨𝐬 𝒙, 𝐬𝐢𝐧 𝒙, 𝒆𝒙 , and 𝐥𝐧 𝒙 Notes
Write your questions
and thoughts here!
Derivatives of 𝐜𝐨𝐬 𝒙 and 𝐬𝐢𝐧 𝒙
𝑑 𝑑
cos 𝑥 sin 𝑥
sinx sin 𝑥 cos 𝑥
cost
𝑑𝑥 𝑑𝑥
Example: Find 𝑓 𝑥 if 𝑓 𝑥 2 sin 𝑥 5 cos 𝑥
Recall:
1117 2005
1 551
Ine1
ln 1 0 ln 0 undet
undefined 𝑒 11 𝑒 𝑎a ln 𝑒 𝑎 tn
O a
9
ax.Ina
Derivatives of Exponential Functions
𝑑 𝑑
𝑎 𝑎 ln 𝑎 𝑒 𝑒 ln 𝑒
ex 𝑒
𝑑𝑥 𝑑𝑥
Example: Find 𝑓 𝑥 if 𝑓 𝑥 2 3𝑒
flxt.at.In2 3ex
Derivatives of Logarithmic Functions
𝑑 1 1 𝑑 𝑑 1 1 1
log 𝑥 ⋅ ln 𝑥 log 𝑥 ⋅
𝑑𝑥 𝑥 ln 𝑥 𝑑𝑥 𝑑𝑥 𝑥 ln 𝑒 𝑥
Example: Find 𝑓 𝑥 if 𝑓 𝑥 log 𝑥 4 ln 𝑥
Find the derivative of each function.
1. 𝑓 𝑥 2 sin 𝑥 5𝑒 2. 𝑓 𝑥 3 4 cos 𝑥 3. 𝑓 𝑥 log 𝑥 sin 𝑥
f x 2cosxtsexflxt 3X.in usinxt'cxt In2 cost
Write your questions Find the value of the derivative at the given point.
and thoughts here!
4. If 𝑓 𝑥 3 ln 𝑥 𝑒 , find 𝑓 5 5. If 𝑓 𝑥 3 cos 𝑥 , find 𝑓 𝜋
3
x
Ite f xk sinx 1 cost
first 3 et t.lt
Eft g.n
s
2.7 Derivatives of 𝐜𝐨𝐬 𝒙, 𝐬𝐢𝐧 𝒙, 𝒆𝒙 , and 𝐥𝐧 𝒙
Calculus
Practice
Find the derivative of each function.
1. 𝑓 𝑥 2 ln 𝑥 2. 𝑓 𝑥 5 3. 𝑓 𝑥 9 cos 𝑥 4. 𝑓 𝑥 5𝑒
5 gsinx sex
5. 𝑓 𝑥 8 ln 𝑥 4 cos 𝑥 𝑒 6. 𝑓 𝑥 15 sin 𝑥 3𝑒 7. 𝑓 𝑥 log 𝑥
8 1 usinxtto 15 cost set
Etusinx 1.12
8. 𝑓 𝑥 log 𝑥 cos 𝑥 9. 𝑓 𝑥 3 3𝑥 𝑥 10. 𝑓 𝑥 3 sin 𝑥
t.int sinx
yx.in 3 32 1 1 3cost
Find the value of the derivative at the given point.
11. If 𝑓 𝑥 3𝑥 6 cos 𝑥, find 𝑓 12. If 𝑓 𝑥 √𝑥 2 ln 𝑥, find 𝑓 4
13. If 𝑓 𝑥 4𝑒 5 sin 𝑥, find 𝑓 0 14. If 𝑓 𝑥 2 cos 𝑥 𝑒 , find 𝑓 𝜋
Find the equation of the tangent line at the given 𝒙-value.
15. 𝑓 𝑥 3 cos 𝑥 𝑥 at 𝑥 16. 𝑓 𝑥 4𝑒 3 sin 𝑥 𝑥 at 𝑥 0
2.7 Derivatives of 𝐜𝐨𝐬 𝒙, 𝐬𝐢𝐧 𝒙, 𝒆𝒙 , and 𝐥𝐧 𝒙 Test Prep
17. What is the slope of the line tangent to the graph of 𝑦 2 ln 𝑥 at the point 𝑥 8?
(A) (B) (C) (D) 16 (E) 4
18. If 𝑥 3, lim
→
(A) ln 3 (B) 1 (C) (D) nonexistent
19. lim
→
(A) 0 (B) 1 (C)
√ (D) (E) nonexistent
20. If 𝑓 𝑥 sin 𝑥, then lim
→
(A) 𝜋 (B) 1 (C) 1 (D) 𝜋
21. Calculator active. Line 𝐿 is tangent to the graph of 𝑦 2 sin 𝑥 at the point 𝑘, 2 sin 𝑘 , where 0 𝑘 2𝜋.
For what value of 𝑘 does the line 𝐿 pass through the origin?
(A) 1.571 (B) 4.493 (C) 4.712 (D) 3.141 (E) There is no such
value of 𝑘.
ANSWER: B
22. 𝑓 𝑥 𝑥 𝑎 cos 𝑥, where 𝑎 is a positive constant and 0 𝑥 2𝜋. Find the 𝑥-coordinates of all points,
0 𝑥 2𝜋, where the line 𝑦 𝑥 𝑎 is tangent to the graph of 𝑓 𝑥 .
