Find the derivative of the function $f(x) = x^{4}(5 \sin x - 3 \cos x)$ with respect to $x$.

Let $f(x) = x^{4}(5 \sin x - 3 \cos x)$.
Using the product rule,$\frac{d}{dx}[u(x)v(x)] = u(x)v'(x) + v(x)u'(x)$:
$f'(x) = x^{4} \frac{d}{dx}(5 \sin x - 3 \cos x) + (5 \sin x - 3 \cos x) \frac{d}{dx}(x^{4})$
$f'(x) = x^{4}(5 \cos x - 3(-\sin x)) + (5 \sin x - 3 \cos x)(4x^{3})$
$f'(x) = x^{4}(5 \cos x + 3 \sin x) + 4x^{3}(5 \sin x - 3 \cos x)$
$f'(x) = 5x^{4} \cos x + 3x^{4} \sin x + 20x^{3} \sin x - 12x^{3} \cos x$
Factoring out $x^{3}$:
$f'(x) = x^{3}(5x \cos x + 3x \sin x + 20 \sin x - 12 \cos x)$