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Derivatives by formula - sin & cos
Last updated at Sept. 6, 2021 by Teachoo
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 sinxc + d cos x Let f (x) = a + b sinxc + d cos x Let u = a + b sin x & v = c + d cos x So, f(x) = 𝑢𝑣 ∴ f’(x) = 𝑢𝑣′ Using quotient rule f’(x) = 𝑢′𝑣 − 𝑣′𝑢 𝑣2 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) = 𝑢𝑣′ = 𝑢′𝑣 − 𝑣′𝑢 𝑣2 = 𝑏 cos𝑥 (𝑐 + 𝑑 cos𝑥) − (− 𝑑 sin𝑥) (𝑎 + 𝑏 sin𝑥) (𝑐 + 𝑑 cos𝑥)2 = 𝑐𝑏 cos𝑥 + 𝑑𝑏 𝑐𝑜𝑠2𝑥 + (𝑑 sin𝑥) (𝑎 + 𝑏 sin𝑥) (𝑐 + 𝑑 cos𝑥)2 = 𝑐𝑏 cos𝑥 + 𝑑𝑏 𝑐𝑜𝑠2𝑥 + 𝑎𝑑 sin𝑥 +𝑏𝑑 sin2𝑥 (𝑐 + 𝑑 cos𝑥)2 = 𝑐𝑏 cos𝑥 + 𝑑𝑏 (𝒄𝒐𝒔𝟐𝒙 + 𝐬𝐢𝐧𝟐𝒙) + 𝑎𝑏 sin𝑥 (𝑐 + 𝑑 cos𝑥)2 = 𝑐𝑏 cos𝑥 + 𝑑𝑏 𝟏+ 𝑎𝑑 sin𝑥 (𝑐 + 𝑑 cos𝑥)2 = 𝑐𝑏 cos𝑥 + 𝑎𝑑 sin𝑥 + 𝑑𝑏 (𝑐 + 𝑑 cos𝑥)2 ∴ f’(x) = 𝒄𝒃 𝒄𝒐𝒔𝒙 + 𝒂𝒅 𝒔𝒊𝒏𝒙 + 𝒅𝒃 (𝒄 + 𝒅 𝒄𝒐𝒔𝒙)𝟐