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Calculus Chain Rule Quiz Q3 2023
Problems 1 – 10. Multiple Choice, calculate the derivatives.
1.
𝑦 = 𝑥 3 tan 𝑥
A. 𝑥 3 sec 2 𝑥
B. 3𝑥 2 sec 2 𝑥
C. 𝑥 3 sec 2 𝑥 + 3𝑥 2 tan 𝑥
D. 3𝑥 2 + sec 2 𝑥
2. 𝑦 =
2𝑥+1
2𝑥−1
A.
−8𝑥
(2𝑥−1)2
B.
−4
(2𝑥−1)2
C.
4𝑥 2
(2𝑥−1)2
D.
8𝑥−4
(2𝑥−1)2
3. Find 𝑔′ (𝑥) if 𝑔(𝑥) = sin3 (4𝑥)
A.
𝑐𝑜𝑠 3 (4𝑥)
B.
3 sin2(4𝑥) cos(4𝑥)
C.
12 sin2 (4𝑥) cos(4𝑥)
D.
4 cos 3 (4𝑥)
4. The equation 5𝑥 − 2.5 = −2(𝑦 + 3) represents the equation of the tangent line to the graph
of 𝑓(𝑥) when 𝑥 = 1. What is the 𝑓 ′ (1)?
5
A. − 2
C.
B. −2
5
D.
none of these
5. If 𝑓(𝑥) = (𝑥 + 1)(𝑥 2 − 2)3 then 𝑓 ′ (𝑥)
A. 6𝑥(𝑥 2 − 2)2
C.
3(𝑥 + 1)(𝑥 2 − 2)2
6. Find 𝑦 ′ if 𝑦 = 𝑒 cot 𝑥
B. 6𝑥(𝑥 + 1)(𝑥 2 − 2)2
D. (𝑥 2 − 2)2 (7𝑥 2 + 6𝑥 − 2)
2
A. −2𝑥 csc 2 (𝑥 2 )𝑒 cot 𝑥
2
2
B.
− csc 2 (𝑥 2 )𝑒 cot 𝑥
D.
−2𝑥 𝑒 csc
A. 2(cos 𝑥 2 − 𝑥 sin 𝑥 2 )
B.
2(cos 𝑥 2 − 2𝑥 2 sin 𝑥 2 )
C. 2𝑥 cos 𝑥 2
D.
−4𝑥 sin 𝑥 2
C.
cot(𝑥 2 )𝑒 cot 𝑥
2 −1
2 (𝑥 2 )
7. Find the second derivative given 𝑔(𝑥) = sin 𝑥 2
𝑑𝑦
8. Find 𝑑𝑥 for 𝑦 = 𝑥 3 √𝑥 + 1
7𝑥 3 +𝑥 2
A. 2
C.
𝑥 2 (7𝑥+6)
B.
√𝑥+1
3𝑥 2
2√𝑥+1
D. 3𝑥 2 √𝑥 + 1
2√𝑥+1
9. Find the slope of the tangent line to the graph of 𝑓(𝑥) = sin(cos 2𝑥) at 𝑥 = 1
A. −2(sin 2)(cos 2)
C. −2(sin 2) (cos(cos 2))
B. −2(sin(cos 2))
D. 2(cos 2)(sin(cos 2))
𝜋
10. Find an equation for the tangent line to the graph of 𝑦 = tan 2𝜃 at 𝜃 = ( , 1)
A. 8𝑥 − 16𝑦 = 𝜋 − 16
B. 8𝑥 + 2𝑦 = 𝜋 + 2
C. 8𝑥 − 4𝑦 = 𝜋 − 4
D. 8𝑥 − 2𝑦 = 𝜋 − 2
8
Free Response
Assume that 𝑓(𝑥) and 𝑔(𝑥) are differentiable functions about which we know very little. In fact,
assume that all we know about these functions is the following table of data:
𝑥
𝑓(𝑥)
𝑓 ′ (𝑥)
𝑔(𝑥)
𝑔′ (𝑥)
0
4
√2
𝜋
2
−1
1
8
1
2
3
−3
3
3
2𝜋
−4
5
Show set up for partial credit.
11. ℎ(𝑥) = 𝑓(𝑥) ∙ 𝑔(𝑥). What is ℎ′ (3)?
12. 𝑘(𝑥) =
𝑓(𝑥)
𝑔(𝑥)
. What is 𝑘 ′ (1) ?
13. 𝑚(𝑥) = 𝑓(𝑔(𝑥)) . Find 𝑚′ (1).
14. 𝑛(𝑥) = √𝑓(𝑥) . Find 𝑛′ (1).
15. 𝑤(𝑥) = 𝑔(𝑥)𝑠𝑖𝑛 𝑥 . What is 𝑤 ′ (0) ?
16. 𝑢(𝑥) =
17. lim
ℎ→0
𝑔(3+ℎ)−𝑔(3)
ℎ
=
𝑡𝑎𝑛 𝜋𝑥
𝑔(𝑥)
. What is 𝑢′ (1)
18. 𝑞(𝑥) = 𝑒 𝑓(𝑥) . What is 𝑞 ′ (0) ?
19. Find the average rate of change in 𝑔(𝑥) on the interval 1 ≤ 𝑥 ≤ 3
20. What is the equation of the tangent line to 𝑓(𝑥) at 𝑥 = 3?