# Example 17

Last updated at Dec. 8, 2016 by Teachoo

Last updated at Dec. 8, 2016 by Teachoo

Transcript

Example 17 Compute the derivative tan x. Let f(x) = tan x We need to find f’ (x) We know that f’(x) = limℎ→0 f 𝑥 + ℎ − f (x)ℎ Here, f(x) = tan x So, f(x + ℎ) = tan (x + ℎ) Putting values f’ (x) = limℎ→0 tan 𝑥 + ℎ − tan𝑥ℎ f’ (x) = limℎ→0 1ℎ ( tan (x + h) – tan x) = limℎ→0 1ℎ sin 𝑥 + ℎ cos 𝑥 + ℎ − sin𝑥 cos𝑥 = limℎ→0 1ℎ cos x sin 𝑥 + ℎ − cos 𝑥 + ℎ sin𝑥 cos 𝑥 + ℎ cos𝑥 = limℎ→0 1ℎ 𝒔𝒊𝒏 𝒙 + 𝒉 𝒄𝒐𝒔𝒙 − 𝒄𝒐𝒔 𝒙 + 𝒉. 𝒔𝒊𝒏𝒙 cos 𝑥 + ℎ cos𝑥 = limℎ→0 1ℎ 𝐬𝐢𝐧 𝒙 + 𝒉 − 𝒙 cos 𝑥 + ℎ cos𝑥 = limℎ→0 1ℎ ( sin 𝑥 + ℎ − 𝑥 ) cos 𝑥 + ℎ cos𝑥 = limℎ→0 1ℎ ( sin ℎ) cos 𝑥 + ℎ cos𝑥 = limℎ→0 sinℎℎ 1 cos 𝑥 + ℎ cos𝑥 = 𝐥𝐢𝐦𝒉→𝟎 𝒔𝒊𝒏𝒉𝒉 × limℎ→0 1 cos 𝑥 + ℎ cos𝑥 = 1 × limℎ→0 1 cos 𝑥 + ℎ cos𝑥 = limℎ→0 1 cos 𝑥 + ℎ cos𝑥 Putting ℎ = 0 = 1 cos 𝑥 + 0 cos𝑥 = 1 cos x . cos𝑥 = 1 𝑐𝑜𝑠2𝑥 = sec2x Hence , f’(x) = sec2x

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