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  1. Chapter 13 Class 11 Limits and Derivatives
  2. Serial order wise

Transcript

Example 20 Find the derivative of f(x) from the first principle, where f(x) is (i) sin x + cos x Given f (x) = sin x + cos x We need to find Derivative of f(x) We know that f’(x) = lim┬(hβ†’0) 𝑓⁑〖(π‘₯ + β„Ž) βˆ’ 𝑓(π‘₯)γ€—/β„Ž Here, f (x) = sin x + cos x f (x + h) = sin (x + h) + cos (x + h) Putting values f’(x) = lim┬(hβ†’0)⁑〖(sin⁑〖(π‘₯ + β„Ž)γ€— + cos⁑(π‘₯ + β„Ž) βˆ’ (sin⁑π‘₯ + cos⁑〖π‘₯)γ€—)/β„Žγ€— Using sin (A + B) = sin A cos B + cos A sin B & cos (A + B) = cos A cos B – sin A sin B = lim┬(hβ†’0)⁑〖sin⁑〖π‘₯ cosβ‘γ€–β„Ž +γ€– cos〗⁑〖π‘₯ sinβ‘γ€–β„Ž + cos⁑〖π‘₯ cosβ‘γ€–β„Ž βˆ’ sin⁑〖π‘₯ γ€– sinγ€—β‘γ€–β„Ž βˆ’γ€– sin〗⁑〖π‘₯ βˆ’γ€– cos〗⁑π‘₯ γ€— γ€— γ€— γ€— γ€— γ€— γ€— γ€— γ€—/hγ€— = lim┬(hβ†’0)⁑〖cos⁑〖π‘₯ sinβ‘γ€–β„Ž βˆ’γ€– sin〗⁑〖π‘₯ sinβ‘γ€–β„Ž + sin⁑〖π‘₯ cosβ‘γ€–β„Ž βˆ’ sin⁑〖π‘₯ +γ€– cos〗⁑〖π‘₯ cosβ‘γ€–β„Ž βˆ’γ€– cos〗⁑π‘₯ γ€— γ€— γ€— γ€— γ€— γ€— γ€— γ€— γ€—/hγ€— = lim┬(hβ†’0)⁑〖sinβ‘γ€–β„Ž γ€–(cos〗⁑〖π‘₯ βˆ’ sin⁑〖π‘₯) + sin⁑〖π‘₯ (cosβ‘γ€–β„Ž βˆ’ 1) + cos⁑π‘₯ (cosβ‘γ€–β„Ž βˆ’ 1)γ€— γ€— γ€— γ€— γ€— γ€—/hγ€— = lim┬(hβ†’0)⁑(sinβ‘γ€–β„Ž γ€–(cos〗⁑〖π‘₯ βˆ’ sin⁑〖π‘₯)γ€— γ€— γ€—/h+sin⁑〖π‘₯ (cosβ‘γ€–β„Ž βˆ’ 1)γ€— γ€—/h+cos⁑〖π‘₯ (cosβ‘γ€–β„Ž βˆ’ 1)γ€— γ€—/β„Ž)" " = lim┬(hβ†’0)⁑〖sinβ‘γ€–β„Ž γ€–(cos〗⁑〖π‘₯ βˆ’γ€– sin〗⁑〖π‘₯)γ€— γ€— γ€—/h+lim┬(hβ†’0) sin⁑〖π‘₯ (cosβ‘γ€–β„Ž βˆ’ 1)γ€— γ€—/h+lim┬(hβ†’0) cos⁑〖π‘₯ (cosβ‘γ€–β„Ž βˆ’ 1)γ€— γ€—/β„Žγ€— = lim┬(hβ†’0)⁑〖"(cos x – sin x)" sinβ‘β„Ž/β„Ž+lim┬(hβ†’0) "(– sin x)" ((1 βˆ’ cosβ‘γ€–β„Ž)γ€—)/β„Žγ€—+lim┬(hβ†’0) "(– cos x)" ((1 βˆ’ cosβ‘γ€–β„Ž)γ€—)/β„Ž = "(cos x – sin x)" (π₯𝐒𝐦)┬(π‘β†’πŸŽ)⁑〖𝐬𝐒𝐧⁑𝒉/π’‰βˆ’sin⁑〖π‘₯ (π₯𝐒𝐦)┬(π‘β†’πŸŽ) γ€— ((𝟏 βˆ’ 𝒄𝒐𝒔⁑〖𝒉)γ€—)/π’‰βˆ’cos⁑π‘₯ (π₯𝐒𝐦)┬(π‘β†’πŸŽ) ((𝟏 βˆ’ 𝒄𝒐𝒔⁑〖𝒉)γ€—)/𝒉〗Using (π‘™π‘–π‘š)┬(β„Žβ†’0) π‘ π‘–π‘›β‘β„Ž/β„Ž = 1 & (π‘™π‘–π‘š)┬(β„Žβ†’0) γ€–(1 βˆ’ π‘π‘œπ‘ γ€—β‘γ€–β„Ž)γ€—/β„Ž = 0 Using (π‘™π‘–π‘š)┬(β„Žβ†’0) π‘ π‘–π‘›β‘β„Ž/β„Ž = 1 & (π‘™π‘–π‘š)┬(β„Žβ†’0) γ€–(1 βˆ’ π‘π‘œπ‘ γ€—β‘γ€–β„Ž)γ€—/β„Ž = 0

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Davneet Singh
Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He has been teaching from the past 9 years. He provides courses for Maths and Science at Teachoo.