Medium
Find the derivative of the following function (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): $(ax^{2} + \sin x)(p + q \cos x)$
Let $f(x) = (ax^{2} + \sin x)(p + q \cos x)$.
Using the product rule,$\frac{d}{dx}[u(x)v(x)] = u(x)v'(x) + v(x)u'(x)$.
Here,$u(x) = ax^{2} + \sin x$ and $v(x) = p + q \cos x$.
$f'(x) = (ax^{2} + \sin x) \frac{d}{dx}(p + q \cos x) + (p + q \cos x) \frac{d}{dx}(ax^{2} + \sin x)$.
$f'(x) = (ax^{2} + \sin x)(-q \sin x) + (p + q \cos x)(2ax + \cos x)$.
$f'(x) = -q \sin x(ax^{2} + \sin x) + (p + q \cos x)(2ax + \cos x)$.
Using the product rule,$\frac{d}{dx}[u(x)v(x)] = u(x)v'(x) + v(x)u'(x)$.
Here,$u(x) = ax^{2} + \sin x$ and $v(x) = p + q \cos x$.
$f'(x) = (ax^{2} + \sin x) \frac{d}{dx}(p + q \cos x) + (p + q \cos x) \frac{d}{dx}(ax^{2} + \sin x)$.
$f'(x) = (ax^{2} + \sin x)(-q \sin x) + (p + q \cos x)(2ax + \cos x)$.
$f'(x) = -q \sin x(ax^{2} + \sin x) + (p + q \cos x)(2ax + \cos x)$.
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