Derivatives of a Function - Basic Rules and Examples

Learning about derivatives is crucial in engineering mathematics. This lecture explores the fundamental rules for finding derivatives, including the Constant Rule, Constant Multiple Rule, Sum & Difference Rules, and Power Rule. Through examples and explanations, you will gain a solid understanding of differentiation and how it applies to various functions.

• Derivatives
• Function
• Engineering Mathematics
• Differentiation
• Basic Rules

seif_1 Follow

Uploaded on Feb 24, 2025 | 0 Views

Download Presentation

Please find below an Image/Link to download the presentation.

The content on the website is provided AS IS for your information and personal use only. It may not be sold, licensed, or shared on other websites without obtaining consent from the author.If you encounter any issues during the download, it is possible that the publisher has removed the file from their server.

You are allowed to download the files provided on this website for personal or commercial use, subject to the condition that they are used lawfully. All files are the property of their respective owners.

The content on the website is provided AS IS for your information and personal use only. It may not be sold, licensed, or shared on other websites without obtaining consent from the author.

Presentation Transcript

1. ENGINEERING MATHEMATICS Lecture 4 / Derivatives of a Function Fall Semester 2018 Instructor Instructor Twana Twana A. Hussein A. Hussein MSc . Structural Engineering MSc . Structural Engineering Twana.ahmad@ishik.edu.iq Twana.ahmad@ishik.edu.iq Civil Engineering Dept. Civil Engineering Dept. Ishik University Ishik University

2. Ishik University DERIVATIVESOFA FUNCTION The process for finding the derivative is called differentiation, In this lecture, we will use d/dx notation to signify differentiation as an operation and employ some basic rules in order to find the derivative. 2 .

3. Ishik University DERIVATIVESOFA FUNCTION 1- Constant Rule: The derivative of a constant is zero; that is, for a constant c: ? ??? = ? Where; c can be any real number 3 .

4. Ishik University DERIVATIVESOFA FUNCTION 2- Constant Multiple Rule: The derivative of a constant multiplied by a function is the constant multiplied by the derivative of the original function: ? ?? ? ?? ? ? ? = ? ? ? 4 .

5. Ishik University DERIVATIVESOFA FUNCTION 3- Sum & Difference Rules: The sum and difference rules say, The derivative of the sum of two functions is the sum of the derivatives of the two functions: So we can work out each derivative separately and then add them. ? ??? ? ?(?) = ? ???(?) ? ???(?) Or ? ? ?(?) = ? ? ? ? 5 .

6. Ishik University DERIVATIVESOFA FUNCTION 4- Power Rule: If n is any real number, then ? ? = ?? ? ??? ? = ??? ? OR ? ? = ??? ? 6 .

7. Ishik University DERIVATIVESOFA FUNCTION Example -1: Find the derivative of the function f f ? = ??? ?? + ? Solution: Applying the property of linearity in conjunction with the power rule. ? ??f f ? = ? ?? f f ? = ??? ?? + ? ? ??(9?3) ? ??(3?) +? ??(7) 7 .

8. Ishik University DERIVATIVESOFA FUNCTION ? ??(?3) 3 ? ??(?) + ? ??(7) f x = 9 ? ??(7) f x = 9 (3?3 1) 3 (?1 1) + Notice also that the constant term 7 yielded 0 f x = 9 3?2 3 ?0+ 0 f f ? = ???? ? 8 .

9. Ishik University DERIVATIVESOFA FUNCTION Example -2: Find the derivative of the function f f ? = ??? ??+ ???. Solution: ? ??f f ? = ? ?? f f ? = ??? ??+ ??? ? ??5?2 ? ???3+? ??7?4 = 9 .

10. Ishik University DERIVATIVESOFA FUNCTION = ?? ???2 ? ???3+ ?? ???4 = 5 2?2 1 3?3 1+ 7 4?4 1 = 5 2?1 3?2+ 7 4?3 = ??? ???+ ???? 10 .

11. Ishik University DERIVATIVESOFA FUNCTION Example -3: Find the derivative of the function f f ? =? ? ? ?= ? ? Solution: since, ? ??f f ? = ? ?? f f ? = ? ? 11 .

12. Ishik University DERIVATIVESOFA FUNCTION ? ??f f ? = ? ?? f f ? = ? ? ? ?? ?? ? ? = ? ?? ? ? = = ? ?? 12 .

13. Ishik University DERIVATIVESOFA FUNCTION 5- Product Rule: The derivative of the product of two functions is NOT the product of the functions' derivatives; rather, it is described by the equation below: ? ??? ? ?(?) = ? ? ? ??(? ? ) + ?(?) ? ??(? ? ) 13 .

14. Ishik University DERIVATIVESOFA FUNCTION 5- Product Rule: This rule is used for the following types of problems; When there is a number multiplied by variable, such as ?? When two variables are multiplied, such as ???? When brackets are multiplied, such as ?? ? ? + ? Or variable multiplied by sine, cos, tan, such as ????(??) 1. 2. 3. 4. 14 .

15. Ishik University DERIVATIVESOFA FUNCTION Example-4: Find the derivative of the function ? ? = ?? ? ? + ? Solution: ? ??? ? ?(?) = ? ? ? ??(? ? ) + ?(?) ? ??(? ? ) ? ? = ?? ? ? + ? + ? + ? ?? ? ? ? = ?? ? (?)+ ? + ? (??) 15 .

16. Ishik University DERIVATIVESOFA FUNCTION ? ? = ?? ? (?)+ ? + ? (??) ? ? = ?? ? + ???+ ??? ? ? = ???+ ??? ? 16 .

17. Ishik University DERIVATIVESOFA FUNCTION 6- Quotient Rule: The derivative of the quotient of two functions is NOT the quotient of the functions' derivatives; rather, it is described by the equation below: ? ??? ? ? ? ? ? ? ??? ? ? ?? ?(?) ?(?) = ? ? ? 17 .

18. Ishik University DERIVATIVESOFA FUNCTION Example-5: Find the derivative of the function ? ? ? = ? + ?? Solution: ? ? ? ? ? ? ? ? ? ?? ?(?) ?(?) = ? ? ? =? (?+??) (?) (?+??) ?+??? ? ?? ?(?) ?(?) 18 .

19. Ishik University DERIVATIVESOFA FUNCTION =? (1+2?) (?) (1+2?) 1+2?2 ? ?? ?(?) ?(?) ? ?? ?(?) ?(?) =1 + 2? 2? 1 + 2?2 ? ?? ?(?) ?(?) 1 = 1 + 2?2 19 .

20. Ishik University DERIVATIVESOFA FUNCTION 7- Chain Rule: The chain rule is used to differentiate composite functions. As such, it is a vital tool for differentiating most functions of a certain complexity. It states:. ? ???(? ? ) = ? (? ? ) ? ? 20 .

21. Ishik University DERIVATIVESOFA FUNCTION Example 6: Find the derivative of the function ? ? = (??? ???)? Solution: ? ???(? ? ) = ? (? ? ) ? ? ? ??(??? ???)? ? ????? ??? ? ? = 21 .

22. Ishik University DERIVATIVESOFA FUNCTION ? ?? ? ?? ? ? = (??? ???)? ??? ??? ? ? = ?(??? ???)? ???? ?? ??? ???? ? ? = ? ???? ?? 22 .

23. Ishik University DERIVATIVESOFA FUNCTION Example 7: Find the derivative of the function ? ? ? = ?? + ? Solution: Rule No. 6 ? ??? ? = ? ?? ? ?? + ? ?? + ? ? (?) ?? + ? ?? + ?? ? ? = 23 .

24. Ishik University DERIVATIVESOFA FUNCTION ?? + ? ? (?) ?? + ? ?? + ?? ? ? = ?? + ? (?) (?)(?) ?? + ?? ? ? = ? ? ? = ?? + ?? 24 .

25. Ishik University DERIVATIVESOFA FUNCTION Example 7: Find the derivative of the function ? ? ? = ?? + ? Solution: Rule No. 7 ? ??+?= ?? + ? ? ? ? = ? ?? ? ???? + ? ? ? = ?? + ? ? 25 .

26. Ishik University DERIVATIVESOFA FUNCTION ? ?? ? ???? + ? ? ? = ?? + ? ? ? ? = ? ?? + ? ? ? ? ??+?? ? ? = 26 .

27. Ishik University DERIVATIVESOFA FUNCTION Example 8: Find the derivative of the function ?? ? ? = ?? + ? Solution: Rule No. 6 ? ?? ?? ? ? = ?? + ? ? ? =?? + ? ?? ?? ?? + ? ?? + ?? 27 .

28. Ishik University DERIVATIVESOFA FUNCTION ? ? =?? + ? ?? ?? ?? + ? ?? + ?? ? ? =(?? + ?) (?) ?? ? ?? + ?? ? ? =?? + ? ?? ?? + ?? ? ? ? = 28 ?? + ?? .

29. Ishik University DERIVATIVESOFA FUNCTION Example 8: Find the derivative of the function ?? ? ? = ?? + ? Solution: Rule No. 5 and 7 ?? ??+?= ?? ?? + ? ? ? ? = ? ?? ? ? = ?? ?? + ? ? 29 .

30. Ishik University DERIVATIVESOFA FUNCTION ? ?? ? ? = ?? ?? + ? ? ? ?? ?? + ? ?+ ?? + ? ?? ?? ???? ? ?? ? ?? + ? ?? ?? + ? ? ? ?? + ? + ?? ?? + ??+ 30 .

31. Ishik University DERIVATIVESOFA FUNCTION 4? 4?2+ 16? + 16+ 2 2? + 4 4? 2? + 4 + 2 4?2+ 16? + 16 4?2+ 16? + 16 2? + 4 8?2 16? + 8?2+ 32? + 32 4?2+ 16? + 16 2? + 4 8 2? + 4 ? = 4?2+ 16? + 16 2? + 4 ?? + ?? 31 .