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Basic Cal_7.Rules of Differentiation (Part 2).pdf
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- 3. GOALS FOR THISWEEK I AM ABLE TO: ● derive the differentiation rules; ● apply the differentiation rules in computing the derivatives 1. algebraic functions, 2. exponential functions, 3. logarithmic functions, and 4. trigonometric functions; ● display perseverance to solve for the derivatives of functions. 3
- 4. CHAIN RULE 𝑑 𝑑𝑥 𝑢𝑛 = 𝑛𝑢𝑛−1 𝑑𝑢 𝑑𝑥 EXAMPLE1. Find the derivative of 𝑦 = 𝑥2 + 5 𝑦 = 𝑥2 + 5 1/2
- 5. CHAIN RULE EXAMPLE 1.Find the derivative of 𝑦 = 𝑥2 + 5 𝑦 = 𝑥2 + 5 1/2 Let 𝑢 = 𝑥2 + 5 𝑛 = 1/2 𝑑𝑦 𝑑𝑥 = 1 2 𝑥2 + 5 − 1 2 𝑑 𝑑𝑥 (𝑥2 + 5) 𝑑𝑦 𝑑𝑥 = 1 2 𝑥2 + 5 −1/2 2𝑥 𝑑𝑦 𝑑𝑥 = 𝑥 𝑥2+5 1/2 𝑑𝑦 𝑑𝑥 = 𝑥 𝑥2+5
- 6. CHAIN RULE EXAMPLE 2.Find the derivative of 𝑦 = 𝑥 + 5 2 𝑦 = 𝑥 + 5 2 Let 𝑢 = 𝑥 + 5 𝑛 = 2 𝑑𝑦 𝑑𝑥 = 2 𝑥 + 5 2−1 𝑑 𝑑𝑥 𝑥 + 5 𝑑𝑦 𝑑𝑥 = 2(𝑥 + 5)(1) 𝑑𝑦 𝑑𝑥 = 2𝑥 + 10
- 7. Differentiation of Exponential and LogarithmicFunctions LESSON 7.2 7
- 8. The next setof functions that we like to focus on are exponential and logarithmic functions. The most commonly used exponential form in a calculus course is the natural exponential function 𝒆𝒙 . 8 Summary of Derivative of Exponential Functions
- 9. Solution: 𝑑 𝑑𝑥 = 𝑎𝑥 = 𝑎𝑥 ln𝑎 𝑑 𝑑𝑥 = (𝑥𝟑 + 𝟑𝑥 ) = 𝟑𝑥𝟐 + 𝟑𝑥 (ln 𝟑) EXAMPLE 1. Find the derivative of 𝑦 = 𝑥3 + 3𝑥 using the differentiation rule. Formula Solved the derivative of 𝑥𝟑 and 𝟑𝑥 .
- 10. Solution: 𝑑 𝑑𝑥 𝑒𝑢 = 𝑒𝑢𝑑𝑢 𝑑𝑥 𝑑𝑦 𝑑𝑥 = 𝑒𝑒𝑥 = 𝑒𝑒𝑥 𝑑 𝑑𝑥 𝑒𝑥 = 𝑒𝑒𝑥 ∙ 𝑒 ∙ 𝑑 𝑑𝑥 (𝑥) = 𝑒𝑒𝑥 ∙ 𝑒 ∙ (1) = 𝑒𝑒𝑥+1 EXAMPLE 2. Find the derivative of 𝑦 = 𝑒𝑒𝑥. 10 Let 𝒖 = 𝒆𝒙 𝒅𝒖 = 𝒆 Formula Differentiation is linear. Differentiated them separately and pulled out constant factors Derivative of x is 1
- 11. EXAMPLE 3. Differentiate 𝑦= 𝑥 ∙ ln 𝑥 − 𝑥 11 Solution: 𝑑𝑦 𝑑𝑥 = 𝑥 ∙ ln 𝑥 − 𝑥 = 𝑑 𝑑𝑥 𝑥 ∙ ln 𝑥 − 𝑑 𝑑𝑥 (𝑥) = 𝑑 𝑑𝑥 𝑥 ∙ ln 𝑥 + 𝑥 ∙ 𝑑 𝑑𝑥 ln 𝑥 − 1 = 1 ∙ ln 𝑥 + 𝑥 ∙ 1 𝑥 − 1 = ln 𝑥 Let 𝒖 = 𝒙 𝒅𝒖 = 𝟏 𝒗 = 𝐥𝐧 𝒙 𝒅𝒗 = 𝟏 𝒙 (to be discussed in logarithm) Differentiation is linear. Differentiated them separately and pulled out constant factors Apply the product rule Simplified (derivative of 𝒍𝒏𝒙 is 𝟏 𝒙 ) Final answer
- 12. EXAMPLE 4. Findthe derivative of 𝑦 = 𝑒4𝑥+7. 12 Solution: 𝑑 𝑑𝑥 𝑒𝑢 = 𝑒𝑢 𝑑𝑢 𝑑𝑥 = 𝑒4𝑥+7 𝑑 𝑑𝑥 (4𝑥 + 7) = 𝑒4𝑥+7 4 ∙ 𝑑 𝑑𝑥 𝑥 + 𝑑 𝑑𝑥 (7) = 𝑒4𝑥+7 4 ∙ 1 + 0 = 𝟒𝒆𝟒𝒙+𝟕 Let 𝒖 = 4𝑥 + 7 𝒅𝒖 = 𝟒 Formula Differentiation is linear. Differentiated them separately and pulled out constant factors Differentiated each term Final answer
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- 14. Expressions written inexponential form can be converted to logarithmic function and vice versa. Exponential Form to Logarithmic Form 53 = 125 ⟹ log5 125 = 3 490.5 = 7 ⟹ log49 7 = 0.5 Logarithmic Form to Exponential Form log2 8 = 3 ⟹ 23 = 8 log3 81 = 4 ⟹ 34 = 81 Hence 𝒚 = 𝐥𝐨𝐠𝒃 𝒙 can be written as 𝑏𝑦 = 𝑥 and 𝒚 = 𝐥𝐨𝐠𝒆 𝒙 can be written as 𝑒𝑦 = 𝑥. • natural logarithms are to the base 𝒆 • 𝐥𝐧 𝒙 is used for natural logarithms
- 15. The derivative ofthe Natural Logarithm Function If 𝑦 = ln 𝑥, then 𝑑 𝑑𝑥 ln 𝑥 = 1 𝑥 . 15 If 𝑢 is a differentiable function of 𝑥, then according to the Chain Rule:
- 16. Derivative of LogarithmicFunctions other than the natural logarithms. 𝑑 𝑑𝑥 (log𝑏 𝑥) = 1 𝑥 ln 𝑏 . 16 If 𝑢 is a differentiable function of 𝑥, then
- 17. EXAMPLE 1. Differentiate𝑦 = ln 5𝑥 17 Solution: Use 𝑑 𝑑𝑥 ln 𝑢 = 1 𝑢 ∙ 𝑑𝑢 𝑑𝑥 𝑦 = ln(5𝑥) 𝑑𝑦 𝑑𝑥 = 1 5𝑥 ∙ 𝑑 𝑑𝑥 (5𝑥) 𝑑𝑦 𝑑𝑥 = 1 5𝑥 5 𝑑𝑦 𝑑𝑥 = 5 5𝑥 = 1 𝑥 Let 𝒖 = 𝟓𝒙 𝒅𝒖 = 𝟓
- 18. EXAMPLE 1. Differentiate𝑦 = ln 5𝑥 18 𝑦 = ln 5𝑥 𝑑𝑦 𝑑𝑥 = 1 𝑥
- 19. EXAMPLE 2.Find thederivative of 𝑦 = ln(𝑥3 + 4). 19 Solution: Use 𝑑 𝑑𝑥 ln 𝑢 = 1 𝑢 ∙ 𝑑𝑢 𝑑𝑥 𝑦 = ln(𝑥3 + 4) 𝑑𝑦 𝑑𝑥 = 1 𝑥3+4 ∙ 𝑑 𝑑𝑥 (𝑥3 + 4) 𝑑𝑦 𝑑𝑥 = 1 𝑥3+4 3𝑥2 𝑑𝑦 𝑑𝑥 = 3𝑥2 𝑥3+4 Let 𝒖 = 𝑥3 + 4 𝒅𝒖 = 𝟑𝒙𝟐
- 20. 20 𝑑𝑦 𝑑𝑥 = 3𝑥2 𝑥3 + 4 𝑦= ln 5𝑥 EXAMPLE 2.Find the derivative of 𝑦 = ln(𝑥3 + 4).
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- 22. EXAMPLE. Differentiate 𝑦 =sin 4𝑥 Chain rule has been applied here. Differentiation is linear. Differentiated them separately and pulled out constant factor Final answer
- 23. EXAMPLE. Differentiate 𝑦 =cos(2𝑥) Chain rule has been applied here. Differentiation is linear. Differentiated them separately and pulled out constant factor Derivative of 𝒙 is 1. Final answer
- 24. EXAMPLE. Differentiate 𝑦 =tan(2𝑥) Chain rule has been applied here. Differentiation is linear. Differentiated them separately and pulled out constant factor Derivative of 𝒙 is 1. Final answer
- 25. EXAMPLE. Differentiate 𝑦 =4𝑥2 +cot 𝑥 Differentiation is linear. Differentiated them separately and pulled out constant factor. Applied the differentiation rule for 𝑐𝑜𝑡 𝑥 and applied the power rule for 𝑥𝟐. Final answer
- 26. EXAMPLE. Differentiate 𝑦 =sec(2𝑥) Differentiation is linear. Differentiated them separately and pulled out constant factor. Differentiated them separately and pulled out constant factor. Derivative of x is 1. Final answer
- 27. EXAMPLE. Differentiate 𝑦 =csc 5𝑥 Differentiation is linear. Differentiated them separately and pulled out constant factor. Differentiated them separately and pulled out constant factor. Derivative of x is 1. Final answer
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- 29. Derivative of Exponentialand Logarithmic Functions Exponential Functions Logarithmic Functions 𝒅 𝒅𝒙 𝒍𝒏 𝒙 = 𝟏 𝒙
- 30. Rules of Derivativesof Trigonometric Functions RULE 1 RULE 2 RULE 3
- 31. Rules of Derivativesof Trigonometric Functions RULE 4 RULE 5 RULE 6
- 32. 32 NICE job for LISTENING WELL! Askme question/s… Message me @: ❑ messenger: Jessel-Ann Orencio Lpt ❑ email: jessel-ann.orencio@deped.gov.ph
- 33. REFERENCES ❑ Basic Calculus– Grade 11 Alternative Delivery Mode Quarter 3 – Module 7: Rules of Differentiation, First Edition, 2020 Department of Education ❑ Department of Education (2016). Teaching Guide for Senior High School Basic Calculus ❑ Differential Calculus Technological University of the Philippines
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