Miscellaneous

Chapter 12 Class 11 Limits and Derivatives
Serial order wise

Transcript

Misc 20 Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers): a + b sin﷮x﷯﷮c + d cos x﷯ Let f (x) = a + b sin﷮x﷯﷮c + d cos x﷯ Let u = a + b sin x & v = c + d cos x So, f(x) = 𝑢﷮𝑣﷯﷯ ∴ f’(x) = 𝑢﷮𝑣﷯﷯﷮′﷯ Using quotient rule f’(x) = 𝑢﷮′﷯𝑣 − 𝑣﷮′﷯𝑢﷮ 𝑣﷮2﷯﷯ Finding u’ & v’ u = a + b sin x u’ = (a + b sin x)’ = 0 + b cos x = b cos x & v = c + d cos x v’= 0 + d (– sin x) = – d sin x f’(x) = 𝑢﷮𝑣﷯﷯﷮′﷯ = 𝑢﷮′﷯𝑣 − 𝑣﷮′﷯𝑢﷮ 𝑣﷮2﷯﷯ = 𝑏 cos﷮𝑥 (𝑐 + 𝑑 cos﷮𝑥) − (− 𝑑 sin﷮𝑥) (𝑎 + 𝑏 sin﷮𝑥)﷯﷯﷯﷯﷮ (𝑐 + 𝑑 cos﷮𝑥﷯)﷮2﷯﷯ = 𝑐𝑏 cos﷮𝑥 + 𝑑𝑏 𝑐𝑜𝑠2𝑥 + (𝑑 sin﷮𝑥) (𝑎 + 𝑏 sin﷮𝑥)﷯﷯﷯﷮ (𝑐 + 𝑑 cos﷮𝑥﷯)﷮2﷯﷯ = 𝑐𝑏 cos﷮𝑥 + 𝑑𝑏 𝑐𝑜𝑠2𝑥 + 𝑎𝑑 sin﷮𝑥 +𝑏𝑑 sin2﷮𝑥﷯﷯﷯﷮ (𝑐 + 𝑑 cos﷮𝑥﷯)﷮2﷯﷯ = 𝑐𝑏 cos﷮𝑥 + 𝑑𝑏 (𝒄𝒐𝒔𝟐𝒙 + 𝐬𝐢𝐧𝟐﷮𝒙﷯) + 𝑎𝑏 sin﷮𝑥﷯﷯﷮ (𝑐 + 𝑑 cos﷮𝑥﷯)﷮2﷯﷯ = 𝑐𝑏 cos﷮𝑥 + 𝑑𝑏 𝟏﷯+ 𝑎𝑑 sin﷮𝑥﷯ ﷯﷮ (𝑐 + 𝑑 cos﷮𝑥﷯)﷮2﷯﷯ = 𝑐𝑏 cos﷮𝑥 + 𝑎𝑑 sin﷮𝑥 ﷯+ 𝑑𝑏 ﷯﷮ (𝑐 + 𝑑 cos﷮𝑥﷯)﷮2﷯﷯ ∴ f’(x) = 𝒄𝒃 𝒄𝒐𝒔﷮𝒙 + 𝒂𝒅 𝒔𝒊𝒏﷮𝒙 ﷯+ 𝒅𝒃 ﷯﷮ (𝒄 + 𝒅 𝒄𝒐𝒔﷮𝒙﷯)﷮𝟐﷯﷯