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Ex 13.2, 4 - Find derivative from first principle (i) x3 - 27

Ex 13.2, 4 - Chapter 13 Class 11 Limits and Derivatives - Part 2
Ex 13.2, 4 - Chapter 13 Class 11 Limits and Derivatives - Part 3


Transcript

Ex 13.2, 4 Find the derivative of the following functions from first principle. (i) x3 – 27 Let f(x) = x3 – 27 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 f (x) = x3 – 27 f (x + h) = (x + h)3 – 27 Putting values f’(x) = lim┬(h→0)⁡〖(((x + h)3 − 27) − (x3 − 27))/h〗 = lim┬(h→0)⁡〖((x + h)3 − 27− x3 + 27)/h〗 = lim┬(h→0)⁡〖((x + h)3 − x3 − 27 + 27)/h〗 = lim┬(h→0)⁡〖((x + h)3 − x3 )/h〗 = (𝑙𝑖𝑚)┬(ℎ→0)⁡〖(𝑥3 + ℎ3 + 3𝑥2 ℎ + 3𝑥ℎ2 − 𝑥3)/ℎ〗 = (𝑙𝑖𝑚)┬(ℎ→0)⁡〖(ℎ3 + 3𝑥2 ℎ + 3𝑥ℎ2 − 𝑥3 + 𝑥3)/ℎ〗 = (𝑙𝑖𝑚)┬(ℎ→0)⁡〖(ℎ ( ℎ2 +3𝑥2 + 3𝑥ℎ) )/ℎ〗 = lim┬(h→0)⁡〖ℎ2+3𝑥2+3𝑥ℎ〗 Putting h = 0 = (0)2 + 3x2 + 3x(0) = 0 + 3x2 + 0 = 3x2 Hence, f’(x) = 3x2

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Davneet Singh has done his B.Tech from Indian Institute of Technology, Kanpur. He has been teaching from the past 12 years. He provides courses for Maths, Science, Social Science, Physics, Chemistry, Computer Science at Teachoo.