# Misc 19 - Chapter 5 Class 12 Continuity and Differentiability

Last updated at March 12, 2021 by Teachoo

Last updated at March 12, 2021 by Teachoo

Transcript

Misc 19 Using mathematical induction prove that ๐/๐๐ฅ(๐ฅ^๐) = ใ๐๐ฅใ^(๐โ1) for all positive integers ๐. Let ๐(๐) : ๐/๐๐ฅ (๐ฅ^๐) = ใ๐๐ฅใ^(๐โ1) For ๐ = ๐ Solving LHS (๐(๐ฅ^1)" " )/๐๐ฅ = ๐๐ฅ/๐๐ฅ = 1 = RHS Thus, ๐ท(๐) is true for ๐ = 1 Let us assume that ๐ท(๐) is true for ๐โ๐ต ๐ท(๐) : (๐ (๐ฅ^๐))/๐๐ฅ = ใ๐ ๐ฅใ^(๐โ1) Now We have to prove that P(๐+๐) is true ๐(๐+1) : (๐(๐ฅ^(๐ + 1))" " )/๐๐ฅ = ใ(๐+1) ๐ฅใ^(๐ + 1 โ 1) (๐(๐ฅ^(๐ + 1)))/๐๐ฅ = ใ(๐+1) ๐ฅใ^๐ Taking L.H.S (๐(๐ฅ^(๐ + 1)))/๐๐ฅ = (๐(๐ฅ^(๐ ). ๐ฅ))/๐๐ฅ Using product rule As (๐ข๐ฃ)โ = ๐ขโ๐ฃ + ๐ฃโ๐ข where u = xk & v = x = (๐(๐ฅ^๐)" " )/๐๐ฅ . ๐ฅ + ๐(๐ฅ )/๐๐ฅ . ๐ฅ^(๐ ) = (๐ (๐^๐)" " )/๐ ๐ . ๐ฅ + 1 . ๐ฅ^(๐ ) = (ใ๐. ๐ใ^(๐โ๐) ) . ๐ฅ+๐ฅ^๐ = ใ๐. ๐ฅใ^(๐โ1 + 1) .+๐ฅ^๐ = ใ๐. ๐ฅใ^๐+๐ฅ^๐ = ๐ฅ^๐ (๐+1) = R.H.S Hence proved (From (1): (๐(๐ฅ^๐ ") " )/๐๐ฅ = ใ๐ ๐ฅใ^(๐โ1) ) Thus , ๐ท(๐+๐) is true when ๐ท(๐) is true Therefore, By Principle of Mathematical Induction ๐(๐) : ๐/๐๐ฅ (๐ฅ^๐) = ใ๐๐ฅใ^(๐โ1) is true for all ๐โ๐ต

Miscellaneous

Misc 1

Misc 2

Misc 3

Misc 4

Misc 5 Important

Misc 6 Important

Misc 7 Important

Misc 8

Misc 9 Important

Misc 10

Misc 11 Important

Misc 12

Misc 13 Important

Misc 14 Important

Misc 15 Important

Misc 16 Important

Misc 17 Important

Misc 18

Misc 19 Important You are here

Misc 20

Misc 21

Misc 22

Misc 23 Important

About the Author

Davneet Singh

Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He has been teaching from the past 10 years. He provides courses for Maths and Science at Teachoo.