ECE 598: The Speech Chain Lecture 1: Dimensional Analysis; Cosines; Logs and Exponentials.
Derivatives)and)Integrals)of)Logs)and)Exponentials) · 2019. 2. 4. · LessonC&...
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Lesson #67 Derivatives and Integrals of Logs and Exponentials ln = 1 ! = 1 ln ! = ! ! = ! ln () 1 = + ! = ! + ! = ! ln () + Example #1: If = ( ! ), find !" !" . Example #2: = ln ( )
Transcript of Derivatives)and)Integrals)of)Logs)and)Exponentials) · 2019. 2. 4. · LessonC&...
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Lesson #67
Derivatives and Integrals of Logs and Exponentials
𝑑𝑑𝑥
ln 𝑥 =1𝑥
𝑑𝑑𝑥
𝑙𝑜𝑔!𝑥 =1
𝑥 ln 𝑏
𝑑𝑑𝑥
𝑒! = 𝑒! 𝑑𝑑𝑥
𝑏! = 𝑏!ln (𝑏)
1𝑥𝑑𝑥 = 𝑙𝑛 𝑥 + 𝑐
𝑒! 𝑑𝑥 = 𝑒! + 𝑐
𝑏! 𝑑𝑥 =𝑏!
ln (𝑏)+ 𝑐
Example #1: If 𝑦 = 𝑙𝑛(𝑥!), find !"
!".
Example #2: 𝑦 = ln (𝑙𝑛𝑥)
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Lesson #67
Example #3: 𝑦 = 𝑥! ∙ ln (𝑥) You try: 𝑦 = 𝑠𝑖𝑛𝑥 ∙ 𝑙𝑛𝑥 𝑦 = 𝑠𝑖𝑛!(𝑙𝑛𝑥) Example #4: 𝑦 = 𝑙𝑜𝑔!(𝑐𝑜𝑠𝑥) You try: 𝑦 = 2𝑙𝑜𝑔! 𝑥
Example #5: 𝑦 = !"#!!!"#
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Lesson #67
You try: 𝑦 = !"#$!"#
Example #6: 𝑑𝑑𝑥
𝑙𝑛sin (𝑥)2 − 5𝑥!
Example #7: Use logarithmic differentiation to find 𝒅𝒚
𝒅𝒙.
𝑦 = 𝑥 1 + 𝑥! Example #8: Find 𝒅𝒚
𝒅𝒙 by logarithmic differentiation.
𝑦 = 𝑠𝑖𝑛 𝑥 𝑡𝑎𝑛!(𝑥)
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Lesson #67
Example #9: 5𝑥!
𝑥! − 6𝑑𝑥
You try:
5𝑥!
𝑥! + 1𝑑𝑥
3𝑥𝑥! + 9
𝑑𝑥
Example #10:
12 − 𝑥
𝑑𝑥!
!
You try:
𝑥𝑥! + 8
𝑑𝑥!
!
