Let y=(1)/(x)sin(x). (dy)/(dx)=

Identify function: Identify the function to differentiate: y = (1/x)sin(x).Recognize product: Recognize that the function is a product of two functions, (1/x) and sin(x).Apply product rule: Apply the product rule for differentiation: (d/dx)[u(x)v(x)] = u'(x)v(x) + u(x)v'(x), where u(x) = 1/x and v(x) = sin(x).Differentiate u(x): Differentiate u(x) = 1/x to get u'(x) = -1/x^2.Differentiate v(x): Differentiate v(x) = sin(x) to get v'(x) = cos(x).Substitute into formula: Substitute u'(x) and v'(x) into the product rule formula: (dy/dx) = (-1/x^2)sin(x) + (1/x)cos(x).Simplify expression: Simplify the expression: (dy/dx) = -sin(x)/x^2 + cos(x)/x.

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Let $y=\frac{1}{x} \sin (x)$ . $\newline$ $\frac{d y}{d x}=$

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Q. Let

y=\frac{1}{x} \sin (x)

\newline

\frac{d y}{d x}=

Identify function: Identify the function to differentiate: $y = \frac{1}{x}\sin(x)$ .
Recognize product: Recognize that the function is a product of two functions, $(\frac{1}{x})$ and $\sin(x)$ .
Apply product rule: Apply the product rule for differentiation: $(\frac{d}{dx})[u(x)v(x)] = u'(x)v(x) + u(x)v'(x)$ , where $u(x) = \frac{1}{x}$ and $v(x) = \sin(x)$ .
Differentiate $u(x)$ : Differentiate $u(x) = \frac{1}{x}$ to get $u'(x) = -\frac{1}{x^2}$ .
Differentiate $v(x)$ : Differentiate $v(x) = \sin(x)$ to get $v'(x) = \cos(x)$ .
Substitute into formula: Substitute $u'(x)$ and $v'(x)$ into the product rule formula: $(\frac{dy}{dx}) = (-\frac{1}{x^2})\sin(x) + (\frac{1}{x})\cos(x)$ .
Simplify expression: Simplify the expression: $(\frac{dy}{dx}) = -\frac{\sin(x)}{x^\(2$} + \frac{\cos(x)}{x}\.