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 Example 19 - Find derivative from first principle - Class 11 - Examples

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  1. Chapter 13 Class 11 Limits and Derivatives
  2. Serial order wise
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Example 19 Find the derivative of f from the first principle, where f is given by (i) f(x) = 2x + 3﷮x − 2﷯ Let f (x) = 2x + 3﷮x − 2﷯ We need to find Derivative of f(x) i.e. f’ (x) We know that f’(x) = lim﷮h→0﷯ f﷮ x + h﷯ − f(x)﷯﷮h﷯ Here, f (x) = 2x + 3﷮x − 2﷯ So, f (x + h) = 2 x + h﷯ + 3﷮ x + h﷯− 2﷯ Putting values f’(x) = lim﷮h→0﷯﷮ 2 𝑥 + ℎ﷯+3﷮ 𝑥 + ℎ﷯− 2﷯﷯ − 2𝑥 + 3﷮𝑥 − 2 ﷯﷯﷮ℎ﷯﷯ = lim﷮h→0﷯﷮ 𝑥 − 2﷯ 2 𝑥 + ℎ﷯ + 3﷯− 𝑥 + ℎ − 2﷯ (2𝑥 + 3)﷮ 𝑥 + ℎ −2﷯ (𝑥 − 2)﷯ ﷮ℎ﷯﷯ = lim﷮h→0﷯﷮ 𝑥 −2﷯ 2𝑥 +2ℎ + 3﷯ − 𝑥 + ℎ −2﷯ 2𝑥 +3﷯﷮ℎ 𝑥 + ℎ − 2 ﷯ (𝑥 − 2)﷯﷯ = lim﷮h→0﷯﷮ 𝑥 −2﷯ 2𝑥 + 3﷯ + 2ℎ﷯ − (𝑥 −2)+ ℎ﷯ 2𝑥 + 3﷯ ﷮ℎ 𝑥 + ℎ − 2 ﷯ (𝑥 − 2)﷯﷯ = lim﷮h→0﷯﷮ 𝑥 − 2﷯ 2𝑥 + 3﷯ + 𝑥 −2﷯2ℎ− 𝑥 − 2﷯ 2𝑥 + 3﷯ − ℎ (2𝑥 + 3) ﷮ℎ 𝑥 + ℎ − 2 ﷯ (𝑥 − 2)﷯﷯ = lim﷮h→0﷯﷮ 2ℎ 𝑥 − 2﷯ − ℎ 2𝑥 + 3﷯ + 𝑥 − 2﷯ 2𝑥 +3﷯ − 𝑥 − 2﷯ (2𝑥 + 3) ﷮ℎ 𝑥 + ℎ − 2 ﷯ (𝑥 − 2)﷯﷯ = lim﷮h→0﷯﷮ h 2 x − 2﷯− 2x + 3﷯﷯ + 0﷮ℎ 𝑥 + ℎ − 2 ﷯ (𝑥 − 2)﷯﷯ = lim﷮h→0﷯ ﷮ 2 𝑥 − 2﷯ − (2𝑥 + 3)﷮ 𝑥 − 2﷯ 𝑥 + ℎ − 2﷯ ﷯﷯ = lim﷮h→0﷯﷮ 2𝑥 − 4 − 2𝑥 − 3﷮ 𝑥 −2﷯ 𝑥 + ℎ − 2﷯ ﷯﷯ = lim﷮h→0﷯﷮ − 7﷮ 𝑥 −2﷯ 𝑥 + ℎ − 2﷯ ﷯﷯ Putting h = 0 = − 7﷮ 𝑥 −2﷯ 𝑥 + 0 − 2﷯ ﷯ = − 7﷮ 𝑥 − 2﷯ (𝑥 − 2)﷯ = − 7﷮ 𝑥 − 2﷯2﷯ Hence, f’(x) = − 𝟕﷮ 𝒙 − 𝟐﷯𝟐﷯ Example 19 Find the derivative of f from the first principle, where f is given by (ii) f(x) = x + 1﷮x﷯ Given f (x) = x + 1﷮x﷯ We need to find Derivative of f(x) i.e. f’ (x) We know that f’(x) = lim﷮h→0﷯ f﷮ x + h﷯ − f(x)﷯﷮h﷯ Here, f (x) = x + 1﷮x﷯ So, f (x + h) = (x + h) + 1﷮x + ℎ﷯ Putting values f’(x) = lim﷮h→0﷯﷮ 𝑥 + ℎ﷯+ 1﷮𝑥 + ℎ﷯﷯ − 𝑥 + 1﷮𝑥﷯﷯﷮h﷯﷯ = lim﷮h→0﷯﷮ 𝑥 + ℎ+ 1﷮𝑥 + ℎ﷯ − 𝑥 − 1﷮𝑥 ﷯﷮h﷯﷯ = lim﷮h→0﷯﷮ ℎ+ 1﷮𝑥 + ℎ﷯ − 1﷮𝑥﷯ + 𝑥 − 𝑥﷮ℎ﷯﷯ = lim﷮h→0﷯﷮ ℎ+ 1﷮𝑥 + ℎ﷯ − 1﷮𝑥﷯﷮ℎ﷯﷯ = lim﷮h→0﷯﷮ ℎ+ 𝑥 − (𝑥 − ℎ)﷮ 𝑥 + ℎ﷯ (𝑥)﷯﷮ℎ﷯﷯ = lim﷮h→0﷯﷮ ℎ + (− ℎ)﷮ 𝑥 + ℎ﷯ 𝑥﷯﷮ℎ﷯﷯ = lim﷮h→0﷯﷮ ℎ 1 − 1﷮ 𝑥 + ℎ﷯ 𝑥﷯﷯﷮ℎ﷯﷯ = lim﷮h→0﷯﷮ 1− 1﷮ 𝑥 + ℎ﷯ 𝑥﷯﷯﷯ = lim﷮h→0﷯﷮ 1− 1﷮𝑥 𝑥 + ℎ﷯ ﷯﷯﷯ Putting h = 0 = 1− 1﷮𝑥 𝑥 + 0﷯ ﷯﷯ = 1− 1﷮𝑥2 ﷯﷯ = 1− 1﷮𝑥2 ﷯ Hence, f’(x) = 1− 1﷮𝑥2 ﷯

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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.
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