Question:
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): $\left(a x^{2}+\sin x\right)(p+q \cos x)$
Solution:
Let $f(x)=\left(a x^{2}+\sin x\right)(p+q \cos x)$
By product rule,
$f^{\prime}(x)=\left(a x^{2}+\sin x\right) \frac{d}{d x}(p+q \cos x)+(p+q \cos x) \frac{d}{d x}\left(a x^{2}+\sin x\right)$
$=\left(a x^{2}+\sin x\right)(-q \sin x)+(p+q \cos x)(2 a x+\cos x)$
$=-q \sin x\left(a x^{2}+\sin x\right)+(p+q \cos x)(2 a x+\cos x)$
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