Example 17 - Chapter 13 Class 11 Limits and Derivatives
Last updated at Nov. 30, 2019 by Teachoo
Last updated at Nov. 30, 2019 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) = lim┬(ℎ→0) f〖(𝑥 + ℎ) − f (x)〗/ℎ Here, f(x) = tan x f(x + ℎ) = tan (x + ℎ) Putting values f’ (x) = lim┬(ℎ→0) tan〖(𝑥 + ℎ) −tan𝑥 〗/ℎ = lim┬(ℎ→0) 1/ℎ ( tan (x + h) – tan x) = lim┬(ℎ→0) 1/ℎ (sin(𝑥 + ℎ)/cos(𝑥 + ℎ) − sin𝑥/cos𝑥 ) = lim┬(ℎ→0) 1/ℎ (〖cos x sin〗〖(𝑥 + ℎ) −〖 cos〗〖(𝑥 + ℎ) sin𝑥 〗 〗/cos〖(𝑥 + ℎ) cos𝑥 〗 ) = lim┬(ℎ→0) 1/ℎ (𝒔𝒊𝒏〖(𝒙 + 𝒉) 𝒄𝒐𝒔〖𝒙 − 𝒄𝒐𝒔〖(𝒙 + 𝒉). 〖 𝒔𝒊𝒏〗𝒙 〗 〗 〗/cos〖(𝑥 + ℎ) cos𝑥 〗 ) Using sin (A – B) = sin A cos B – cos B sin A Here A = x + h & B = x = lim┬(ℎ→0) 1/ℎ (𝐬𝐢𝐧(( 𝒙 + 𝒉 ) − 𝒙)/cos〖(𝑥 + ℎ) cos𝑥 〗 ) = lim┬(ℎ→0) 1/ℎ ((sin〖ℎ) 〗)/cos〖(𝑥 + ℎ) cos𝑥 〗 = lim┬(ℎ→0) sinℎ/ℎ 1/cos〖(𝑥 + ℎ) cos𝑥 〗 = (𝐥𝐢𝐦)┬(𝒉→𝟎) 𝒔𝒊𝒏𝒉/𝒉 " ×" lim┬(ℎ→0) 1/cos〖(𝑥 + ℎ) cos𝑥 〗 = 1 × lim┬(ℎ→0) 1/cos〖(𝑥 + ℎ) cos𝑥 〗 = lim┬(ℎ→0) 1/cos〖(𝑥 + ℎ) cos𝑥 〗 Putting ℎ = 0 = 1/cos〖(𝑥 + 0) cos𝑥 〗 = 1/〖cos x〗〖 .cos𝑥 〗 = 1/(〖𝑐𝑜𝑠〗^2 𝑥) = sec2x Hence , f’(x) = sec2x
Derivatives by formula - other trignometric
Derivatives by formula - other trignometric
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