Misc 24 - Find derivative: (ax2 + sin x) (p + q cos x) - Miscellaneous

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Misc 24 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): (ax2 + sin x)( p + q cos x) Let f (x) = (ax2 + sin x) (p + q cos x) Let u = ax2 + sin x & v = p + q cos x ∴ f(x) = uv So, f’(x) = 𝑢𝑣﷯﷮′﷯ Using product rule 𝑢𝑣﷯﷮′﷯= 𝑢﷮′﷯𝑣+ 𝑣﷮′﷯𝑢 Finding u’ & v’ u = ax2 + sin x u’ = (ax2 + sin x)’ = 2ax + cos x v = p + q cos x v’ = (p + q cos x)’ = 0 + q ( – sin x) = – q sin x Now, f’(x) = 𝑢𝑣﷯′ = 𝑢﷮′﷯𝑣+ 𝑣﷮′﷯𝑢 = (2ax + cos x) (p + q cos x) + ( – q sin x) (ax2 + sin x) = – q sin x (ax2 + sin x) + p + q cos x) (2as + cos x)

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