Question

Find the derivative of : x⁴(5 sin x – 3 cos x)

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Answer 1

$\color{magenta}\large\qquad \qquad \underline{ \pmb{{ \mathbb{ \maltese \: \: \: TO \: \: FIND \: \: \: \maltese }}}}$

$\large\mathfrak{Derivative \: \: of } \\ \bf{x}^{4} (5sinx - 3cosx)$

$\color{orange}\qquad \qquad \underline{ \pmb{{ \mathbb{ \maltese \: \: \: IMPORTANT \: \: POINTS \: \: \: \maltese }}}} \\ \\ \bigstar 1. \: \: \: \underline \mathcal{PRODUCT \: \: RULE} \\ \\ \bf\frac{d}{dx} (uv) = u \frac{d}{dx} v + v \frac{d }{dx}u \\ \\ \bigstar2. \: \: \: \bf\frac{d}{dx} sinx = cosx \\ \\ \bigstar3. \: \: \: \bf \frac{d}{dx} cosx = - sinx$

$\color{red}\large\qquad \qquad \underline{ \pmb{{ \mathbb{ \maltese \: \: \: SOLUTION\: \: \: \maltese }}}}$

$\bf \frac{d}{dx} \: x {}^{4} (5sinx - 3cosx) \\ \\ = \sf {x}^{4} \frac{d}{dx} (5sinx - 3cosx) + \sf (5sinx - 3cosx) \frac{d}{dx} {x}^{4} \\ ( \bf \because product \: rule)$

$= \sf {x}^{4} ( \frac{d}{dx}5sinx - \frac{d}{dx}3cosx) \\ \sf + (5sinx - 3cosx)\frac{d}{dx}{x}^{4} \\ \\ = \sf x {}^{4} \{5cosx - ( - 3sinx) \} \\ + \sf(5sinx - 3cosx) {4x}^{3} \\ \\ = \sf x {}^{4} \{5cosx + 3sinx\} + \sf(5sinx - 3cosx) {4x}^{3} \\ \\ \sf = 5 {x}^{4}cosx + 3 {x}^{4} sinx + 20 {x}^{3} sinx - 12 {x}^{3} cosx\\ \\ \bf = {x}^{3}[5xcosx + 3 {x} sinx + 20 sinx - 12 cosx]$

Which is the required Answer

Answer 2

Solution :

{{See Attachment And Get The Complete Solution}}