Misc 19 - Using mathematical induction prove d/dx (xn) = nxn-1 - Miscellaneous

Slide14.JPG
Slide15.JPG Slide16.JPG

  1. Chapter 5 Class 12 Continuity and Differentiability
  2. Serial order wise
Ask Download

Transcript

Misc 19 Using mathematical induction prove that ๐‘‘๏ทฎ๐‘‘๐‘ฅ๏ทฏ( ๐‘ฅ๏ทฎ๐‘›๏ทฏ) = ๐‘›๐‘ฅ๏ทฎ๐‘›โˆ’1๏ทฏ for all positive integers ๐‘›. Let P ๐‘›๏ทฏ : ๐‘‘๏ทฎ๐‘‘๐‘ฅ๏ทฏ ( ๐‘ฅ๏ทฎ๐‘›๏ทฏ) = ๐‘›๐‘ฅ๏ทฎ๐‘›โˆ’1๏ทฏ For ๐‘› = 1 LHS = ๐‘‘( ๐‘ฅ๏ทฎ1๏ทฏ) ๏ทฎ๐‘‘๐‘ฅ๏ทฏ = ๐‘‘๐‘ฅ๏ทฎ๐‘‘๐‘ฅ๏ทฏ = 1 โˆด LHS = RHS Thus ๐‘ƒ ๐‘›๏ทฏ is true for ๐‘› = 1 Let us assume that Let ๐‘ƒ ๐‘˜๏ทฏ is true for ๐‘˜โˆˆ๐‘ต ๐‘ƒ ๐‘˜๏ทฏ : ๐‘‘( ๐‘ฅ๏ทฎ๐‘˜๏ทฏ) ๏ทฎ๐‘‘๐‘ฅ๏ทฏ = ๐‘˜ ๐‘ฅ๏ทฎ๐‘˜โˆ’1๏ทฏ Now We have to prove that P ๐‘˜+1๏ทฏ is true ๐‘ƒ ๐‘˜+1๏ทฏ : ๐‘‘( ๐‘ฅ๏ทฎ๐‘˜ + 1๏ทฏ) ๏ทฎ๐‘‘๐‘ฅ๏ทฏ = ๐‘˜+1๏ทฏ ๐‘ฅ๏ทฎ๐‘˜ + 1 โˆ’ 1๏ทฏ ๐‘‘( ๐‘ฅ๏ทฎ๐‘˜ + 1๏ทฏ) ๏ทฎ๐‘‘๐‘ฅ๏ทฏ = ๐‘˜+1๏ทฏ ๐‘ฅ๏ทฎ๐‘˜๏ทฏ Taking L.H.S ๐‘‘( ๐‘ฅ๏ทฎ๐‘˜ + 1๏ทฏ) ๏ทฎ๐‘‘๐‘ฅ๏ทฏ = ๐‘‘( ๐‘ฅ๏ทฎ๐‘˜ ๏ทฏ. ๐‘ฅ ) ๏ทฎ๐‘‘๐‘ฅ๏ทฏ = ๐‘‘( ๐‘ฅ๏ทฎ๐‘˜๏ทฏ) ๏ทฎ๐‘‘๐‘ฅ๏ทฏ . ๐‘ฅ + ๐‘‘ ๐‘ฅ ๏ทฏ๏ทฎ๐‘‘๐‘ฅ๏ทฏ . ๐‘ฅ๏ทฎ๐‘˜ ๏ทฏ = ๐’…( ๐’™๏ทฎ๐’Œ๏ทฏ) ๏ทฎ๐’…๐’™๏ทฏ . ๐‘ฅ + 1 . ๐‘ฅ๏ทฎ๐‘˜ ๏ทฏ = ๐’Œ. ๐’™๏ทฎ๐’Œโˆ’๐Ÿ๏ทฏ๏ทฏ . ๐‘ฅ+ ๐‘ฅ๏ทฎ๐‘˜๏ทฏ = ๐‘˜. ๐‘ฅ๏ทฎ๐‘˜โˆ’1 + 1๏ทฏ .+ ๐‘ฅ๏ทฎ๐‘˜๏ทฏ = ๐‘˜. ๐‘ฅ๏ทฎ๐‘˜๏ทฏ+ ๐‘ฅ๏ทฎ๐‘˜๏ทฏ = ๐‘ฅ๏ทฎ๐‘˜๏ทฏ ๐‘˜+1๏ทฏ = R.H.S Hence proved Thus , ๐‘ƒ ๐‘˜+1๏ทฏ is true when ๐‘ƒ ๐‘˜๏ทฏ is true โˆด By principal of mathematical Induction ๐‘ƒ ๐‘›๏ทฏ : ๐‘‘๏ทฎ๐‘‘๐‘ฅ๏ทฏ ( ๐‘ฅ๏ทฎ๐‘›๏ทฏ) = ๐‘›๐‘ฅ๏ทฎ๐‘›โˆ’1๏ทฏ is true , ๐‘›โˆˆ๐‘

About the Author

Davneet Singh's photo - Teacher, Computer Engineer, Marketer
Davneet Singh
Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He provides courses for Mathematics from Class 9 to 12. You can ask questions here.
Jail