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Ex 13.2, 4 - Chapter 13 Class 11 Limits and Derivatives - Part 10

Ex 13.2, 4 - Chapter 13 Class 11 Limits and Derivatives - Part 11
Ex 13.2, 4 - Chapter 13 Class 11 Limits and Derivatives - Part 12

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Ex 12.2, 4 Find the derivative of the following functions from first principle. (iv) (π‘₯ + 1)/(π‘₯ βˆ’ 1) Let f (x) = (π‘₯ + 1)/(π‘₯ βˆ’ 1) We need to find Derivative of f(x) i.e. f’ (x) We know that f’(x) = lim┬(hβ†’0) f⁑〖(x + h) βˆ’ f(x)γ€—/h Here, f (x) = (π‘₯ + 1)/(π‘₯ βˆ’ 1) So, f (x + h) = ((π‘₯ + β„Ž) + 1)/((π‘₯ + β„Ž) βˆ’ 1) Ex 12.2, 4 Find the derivative of the following functions from first principle. (iv) (π‘₯ + 1)/(π‘₯ βˆ’ 1) Let f (x) = (π‘₯ + 1)/(π‘₯ βˆ’ 1) We need to find Derivative of f(x) i.e. f’ (x) We know that f’(x) = lim┬(hβ†’0) f⁑〖(x + h) βˆ’ f(x)γ€—/h Here, f (x) = (π‘₯ + 1)/(π‘₯ βˆ’ 1) So, f (x + h) = ((π‘₯ + β„Ž) + 1)/((π‘₯ + β„Ž) βˆ’ 1) Putting values f’(x) = lim┬(hβ†’0)⁑〖([((π‘₯ + β„Ž) + 1)/(π‘₯ + β„Ž βˆ’ 1)] βˆ’[ (π‘₯ + 1)/(π‘₯ βˆ’ 1)])/hγ€— = lim┬(hβ†’0)⁑〖((π‘₯ + β„Ž + 1)/(π‘₯ + β„Ž βˆ’ 1) βˆ’ (π‘₯ + 1)/(π‘₯ βˆ’ 1))/hγ€— = lim┬(hβ†’0)⁑〖(((π‘₯ βˆ’ 1)(π‘₯ + β„Ž + 1) βˆ’ (π‘₯ + 1)( π‘₯ + β„Ž βˆ’ 1))/(( π‘₯ + β„Ž βˆ’ 1 ) (π‘₯ βˆ’ 1)))/β„Žγ€— = lim┬(hβ†’0)⁑〖((π‘₯ βˆ’ 1) ((π‘₯ + 1) + β„Ž) βˆ’ (π‘₯ + 1)( (π‘₯ βˆ’ 1) + β„Ž))/(β„Ž( π‘₯ + β„Ž βˆ’ 1 ) (π‘₯ βˆ’ 1))γ€— = lim┬(hβ†’0)⁑〖((π‘₯ βˆ’ 1)(π‘₯ + 1) + (π‘₯ βˆ’1)β„Ž βˆ’ (π‘₯ + 1)(π‘₯ βˆ’ 1) βˆ’ (π‘₯ + 1) β„Ž)/(β„Ž( π‘₯ + β„Ž βˆ’ 1 ) (π‘₯ βˆ’ 1))γ€— = lim┬(hβ†’0)⁑〖((π‘₯2 βˆ’ 1) + π‘₯β„Ž βˆ’ (π‘₯2 βˆ’ 1) βˆ’ π‘₯β„Ž βˆ’ β„Ž)/(β„Ž( π‘₯ + β„Ž βˆ’ 1 ) (π‘₯ βˆ’ 1))γ€— = (π‘™π‘–π‘š)┬(β„Žβ†’0)⁑〖(βˆ’ 2β„Ž )/(β„Ž (π‘₯ + β„Ž βˆ’ 1) (π‘₯ βˆ’ 1))γ€— = lim┬(hβ†’0)⁑〖(βˆ’2)/((π‘₯ + β„Ž βˆ’ 1) (π‘₯ βˆ’ 1))γ€— Putting h = 0 = (βˆ’2)/((π‘₯ + 0 βˆ’ 1)(π‘₯ βˆ’ 1)) = (βˆ’2)/((π‘₯ βˆ’ 1) (π‘₯ βˆ’ 1)) = (βˆ’2)/(π‘₯ βˆ’ 1)^2 Hence, f’(x) = (βˆ’πŸ)/(𝒙 βˆ’ 𝟏)^𝟐

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Davneet Singh has done his B.Tech from Indian Institute of Technology, Kanpur. He has been teaching from the past 13 years. He provides courses for Maths, Science, Social Science, Physics, Chemistry, Computer Science at Teachoo.