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): $x^{4}(5 \sin x-3 \cos x)$
Solution:
Let $f(x)=x^{4}(5 \sin x-3 \cos x)$
By product rule,
$f^{\prime}(x)=x^{4} \frac{d}{d x}(5 \sin x-3 \cos x)+(5 \sin x-3 \cos x) \frac{d}{d x}\left(x^{4}\right)$
$=x^{4}\left[5 \frac{d}{d x}(\sin x)-3 \frac{d}{d x}(\cos x)\right]+(5 \sin x-3 \cos x) \frac{d}{d x}\left(x^{+}\right)$
$=x^{4}[5 \cos x-3(-\sin x)]+(5 \sin x-3 \cos x)\left(4 x^{3}\right)$
$=x^{3}[5 x \cos x+3 x \sin x+20 \sin x-12 \cos x]$
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