Scribd
Upload
42 views4 pages

Calc 3.6 Packet

The document provides notes on calculating higher-order derivatives in calculus, including notation for first, second, third, and nth derivatives. It includes practice problems for finding second derivatives and evaluating them at specific points, as well as additional derivative calculations. There are also test prep questions with multiple-choice answers related to derivatives.

Uploaded by

aslihancecen007
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
Available Formats
Download as PDF, TXT or read online on Scribd
42 views4 pages

Calc 3.6 Packet

The document provides notes on calculating higher-order derivatives in calculus, including notation for first, second, third, and nth derivatives. It includes practice problems for finding second derivatives and evaluating them at specific points, as well as additional derivative calculations. There are also test prep questions with multiple-choice answers related to derivatives.

Uploaded by

aslihancecen007
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
Available Formats
Download as PDF, TXT or read online on Scribd
You are on page 1/ 4

Calculus 3.

6 Calculating Higher‐Order Derivatives Notes


Write your questions
and thoughts here!
Notation 1st Derivative 2nd Derivative 3rd Derivative nth Derivative
𝑦 𝑦 𝑦 𝑦 𝑦

𝑓 𝑥 𝑓 𝑥 𝑓 𝑥 𝑓 𝑥 𝑓 𝑥

𝑑𝑦 𝑑 𝑦 𝑑 𝑦 𝑑 𝑦
𝑦
𝑑𝑥 𝑑𝑥 𝑑𝑥 𝑑𝑥

Finding the 2nd derivative


𝑓 𝑥 𝑥 2𝑥 5𝑥 3𝑥 9 𝑦 √𝑥 𝑥

𝑓 𝑥
𝑑𝑦
𝑓 𝑥
𝑑𝑥
𝑓 𝑥
𝑑 𝑦
𝑓 𝑥
𝑑𝑥

2nd Derivative with Implicit Differentiation


Find for sin 𝑦 𝑥 𝑦

3.6 Calculating Higher‐Order Derivatives


Calculus
Practice
𝒅𝟐 𝒚
Find based on the given information.
𝒅𝒙𝟐
1. 𝑦 sin 𝑥 ln 5𝑥 2. 𝑦 𝑒 𝑥 3. 𝑦 sin 𝑥
5. 𝑦 2𝑥 1 6. 3𝑥 7. 𝑥𝑦

8. sin 𝑥 𝑦 2𝑥 9. 𝑒 𝑦 1 10. ln 𝑦 5𝑥 3

Evaluate the 2nd derivative at the given point.


11. If 𝑓 𝑥 3𝑥 4𝑥 , find 𝑓 2 . 12. If 𝑓 𝑥 𝑥 ln 𝑥, find 𝑓 1 .
13. If 𝑓 𝑥 3√𝑥 , find 𝑓 4 . 14. If 3 cos 𝑦 5𝑥, find at 2,

15. If , find at 1, 2 16. If ln 𝑥 𝑒 , find at 𝑒, 1

Find the derivatives of the following.


17. 𝑓 𝑥 3𝑥 4𝑥 5𝑥 18. 𝑦 4√𝑥 19. 𝑦 𝑥

𝑓 𝑥
𝑑𝑦
𝑦
𝑓 𝑥 𝑑𝑥

𝑓 𝑥 𝑦
𝑑 𝑦
𝑓 𝑥 𝑑𝑥
𝑦
𝟏
Given 𝒇 𝒙 𝟑𝒙𝟐 𝒙 𝟐, 𝒈 𝒙 , and 𝒉 𝒙 √𝒙 , find the following.
𝒙𝟑

20. 𝑓′′ 2 21. 𝑔′′′ 3 22. 2ℎ′′ 4


3.6 Calculating Higher‐Order Derivatives Test Prep
23. If 𝑓 𝑥 1 , find 𝑓′′ 40 .

(A) 0.068

(B) 1.350

(C) 5.400

(D) 6.750

(E) 540.000
ANSWER: B

24. A curve given by the equation 𝑥 𝑥𝑦 8 has slope given by . The value of at the point
where 𝑥 2 is

(A) 6 (B) 3 (C) 0 (D) 4 (E) undefined

25. If 𝑦 𝑥𝑒 , then

(A) 𝑒 (B) 𝑒 (C) 𝑥 𝑛 𝑒 (D) 𝑥 𝑒 (E) 𝑥 𝑛 𝑒


ANSWER: C

26. Let 𝑔 be the function given by 𝑔 𝑥 cos 𝑥 sin 𝑥 6. Which of the following statements is true for
𝑦 𝑔 𝑥 ?

(A) 𝑦 6

(B) 𝑔 𝑥 𝑔 𝑥

(C) 𝑔 𝑥 6 𝑔 𝑥

(D) 𝑦 cos 𝑥 sin 𝑥

ANSWER: A

You might also like