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

(dy)/(dx)=

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

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Write a formula for an arithmetic sequence

Full solution

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}\.

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