Ex 13.2, 10 - Find derivative of cos x from first principle - Derivatives by formula - sin & cos

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  1. Chapter 13 Class 11 Limits and Derivatives
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Ex 13.2, 10 Find the derivative of cos x from first principle. Let f (x) = cos x We need to find f’(x) We know that f’(x) = lim﷮h→0﷯ f﷮ x + h﷯ − f(x)﷯﷮h﷯ Here, f (x) = cos x So, f (x + h) = cos (x + h) Putting values, f’ (x) = lim﷮h→0﷯﷮ 𝒄𝒐𝒔 𝒙 + 𝒉﷯ − 𝒄𝒐𝒔﷮𝒙﷯﷮h﷯﷯ Using cos A – cos B = – 2 sin 𝐴 + 𝐵﷮2﷯﷯ sin 𝐴 − 𝐵﷮2﷯﷯ = lim﷮h→0﷯﷮ −𝟐 𝒔𝒊𝒏 𝒙 + 𝒙 + 𝒉﷯﷮𝟐﷯﷯ . 𝒔𝒊𝒏 𝒙 + 𝒉﷯ − 𝒙﷮𝟐﷯﷯﷮h﷯﷯ = lim﷮h→0﷯﷮ −2 𝑠𝑖𝑛 2𝑥 + ℎ﷮2﷯﷯ . 𝑠𝑖𝑛 ℎ﷮2﷯﷯﷮h﷯﷯ = lim﷮h→0﷯﷮−2 sin﷮ 2𝑥 + ℎ﷮2﷯﷯﷯. sin ﷮ ℎ﷮2﷯﷯﷮ℎ﷯﷯ = lim﷮h→0﷯﷮− sin﷮ 2𝑥 + ℎ﷮2﷯﷯﷯. sin ﷮ ℎ﷮2﷯﷯﷮ ℎ﷮2﷯﷯﷯ = lim﷮h→0﷯﷮− sin﷮ 2𝑥 + ℎ﷮2﷯﷯﷯. 𝐥𝐢𝐦﷮𝐡→𝟎﷯ 𝐬𝐢𝐧 ﷮ 𝒉﷮𝟐﷯﷯﷮ 𝒉﷮𝟐﷯﷯﷯ = lim﷮h→0﷯﷮− sin﷮ 2𝑥 + ℎ﷮2﷯﷯﷯.𝟏﷯ = lim﷮h→0﷯﷮− sin﷮ 2𝑥 + ℎ﷮2﷯﷯﷯﷯ Putting h = 0 = − sin﷮ 2𝑥 +0﷮2﷯﷯﷯ = − sin﷮ 2𝑥﷮2﷯﷯﷯ = – sin x ∴ f’ (x) = – sin x

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