Find the derivative of the function x^4 (5 x - 3 x). - Vidyadip

Question

Find the derivative of the function $x^4 (5 \sin x - 3 \cos x).$

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Answer

Let $f(x) = x^{4}(5 \sin x-3 \cos x)$
By product rule of differentiation,we have,
$\mathrm{f}^{\prime}(\mathrm{x})=\mathrm{x}^{4} \frac{\mathrm{d}}{\mathrm{dx}}(5 \sin \mathrm{x}-3 \cos \mathrm{x})$ $+(5 \sin x-3 \cos x) \frac{d}{d x}\left(x^{4}\right)$
$=x^{4} \times\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^{4}\right)$
$=x^{4}[5 \cos x-3(-\sin x)]+(5 \sin x-3 \cos x)\left(4 x^{3}\right)$
$\therefore \mathrm{f}^{\prime}(\mathrm{x})=\mathrm{x}^{3}[5 \mathrm{x} \cos \mathrm{x}+3 \mathrm{x} \sin \mathrm{x}+20 \sin \mathrm{x}-12 \cos \mathrm{x}]$

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