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v) Obtain derivatives of the following functions: (i) x sin x (ii) x+cos x (iii) x/sin x ​

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obtain derivatives of the following functions: 1) x sin x 2) x4 + cos x 3) x/sin​

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Answer:The derivatives of given functions are as follows:1) $\frac{d}{dx} (xsin(x)) = sin(x)+xcos(x)$2) $\frac{d}{dx}(x^{4}+cos(x)) = 4x^{3}-sin(x)$3) $\frac{d}{dx}(x/sin(x)) = \frac{sin(x) - xcos(x)}{sin^{2}(x) }$Explanation:1) $\frac{d}{dx} (xsin(x))$We apply product rule here$\frac{d}{dx} (xsin(x))$ = $(x)\frac{d}{dx} (sin(x))+(sin(x))\frac{d}{dx} (x)$$\frac{d}{dx} (xsin(x)) = (x) (cos(x)) + sin(x)$⇒ $\frac{d}{dx} (xsin(x)) = sin(x)+xcos(x)$2) $\frac{d}{dx}(x^{4}+cos(x))$We apply power rule here$\frac{d}{dx}(x^{4}+cos(x)) = \frac{d}{dx} (x^{4})+\frac{d}{dx}(cos(x))$$\frac{d}{dx}(x^{4}+cos(x)) = 4x^{3} + (-sin(x))\\frac{d}{dx}(x^{4}+cos(x)) = 4x^{3} - sin(x)$⇒ $\frac{d}{dx}(x^{4}+cos(x))=4x^{3} - sin(x)$3) $\frac{d}{dx}(x/sin(x))$We apply quotient rule here[tex]\frac{d}{dx}(x/sin(x)) = \frac{sin(x)\frac{d}{dx}(x)- x \frac{d}{dx}(sin(x))}{sin^{2}(x)}\ \frac{d}{dx}(x/sin…

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