Example 16

Transcript

Example 16 Compute the derivative of sin x. Let f (x) = sin x We need to find f’(x) We know that f’(x) = lim﷮h→0﷯ f﷮ x + h﷯ − f(x)﷯﷮h﷯ Here, f (x) = sin x So, f (x + h) = sin ( x + h) Putting values f’(x) = lim﷮h→0﷯﷮ 𝐬𝐢𝐧﷮ 𝒙 + 𝒉﷯ − 𝐬𝐢𝐧﷮𝒙﷯﷯﷮h﷯﷯ Using sin A – sin B = 2 cos 𝐴 + 𝐵﷮2﷯﷯ . sin 𝐴 − 𝐵﷮2﷯﷯ = lim﷮h→0﷯﷮ 𝟐 𝒄𝒐𝒔 𝒙 + 𝒉 + 𝒙﷮𝟐﷯﷯ . 𝒔𝒊𝒏 𝒙 + 𝒉 − 𝒙﷮𝟐﷯﷯﷮h﷯﷯ = lim﷮h→0﷯﷮ 𝑐𝑜𝑠 2𝑥 + ℎ﷮2﷯﷯ . 𝑠𝑖𝑛 ℎ﷮2﷯﷯﷮h﷯﷯ = lim﷮h→0﷯﷮ cos﷮ 2𝑥 + ℎ﷮2﷯﷯﷯ . sin﷮ ℎ﷮2﷯﷯﷮ ℎ﷮2﷯﷯﷯﷯ = lim﷮h→0﷯﷮ cos﷮ 2𝑥 + ℎ﷯﷮2﷯﷯. 𝐥𝐢𝐦﷮𝐡→𝟎﷯ 𝐬𝐢𝐧 ﷮ 𝒉﷮𝟐﷯﷯﷮ 𝒉﷮𝟐﷯﷯﷯ = lim﷮h→0﷯﷮ cos﷮ 2𝑥 + ℎ﷯﷮2﷯﷯ ﷯ × 1 = lim﷮h→0﷯﷮ cos﷮ 2𝑥 + ℎ﷮2﷯﷯﷯﷯ Putting h = 0 = cos﷮ 2𝑥 +0﷮2﷯﷯﷯ = cos ﷮ 2𝑥﷮2﷯﷯﷯ = cos x ∴ f’ (x) = cos x