(d)/(dx)((5)/(x)ln(x))=

Apply Product Rule: Use the product rule for differentiation, which states that the derivative of a product of two functions is the derivative of the first function times the second function plus the first function times the derivative of the second function.Define u and v: Let u = (5/x) and v = ln(x). Then, find the derivatives u' and v'.Differentiate u: Differentiate u = (5/x). u' = -5/x^2.Differentiate v: Differentiate v = ln(x). v' = 1/x.Use Product Rule Formula: Apply the product rule: (u*v)' = u'v + uv'.Substitute into Formula: Substitute u, u', v, and v' into the product rule formula: (-5/x^2)ln(x) + (5/x)(1/x).Simplify Expression: Simplify the expression: (-5ln(x))/x^2 + 5/x^2.Combine Like Terms: Combine like terms: (-5ln(x) + 5)/x^2.

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$\frac{d}{d x}\left(\frac{5}{x} \ln (x)\right)=$

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Algebra 1

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\frac{d}{d x}\left(\frac{5}{x} \ln (x)\right)=

Apply Product Rule: Use the product rule for differentiation, which states that the derivative of a product of two functions is the derivative of the first function $f'(x)$ times the second function $g(x)$ plus the first function $f(x)$ times the derivative of the second function g'(x) \.}
Define \(u and $v$ : Let $u = \frac{5}{x}$ and $v = \ln(x)$ . Then, find the derivatives $u'$ and $v'$ .
Differentiate $u$ : Differentiate $u = \frac{5}{x}$ . $u' = -\frac{5}{x^2}$ .
Differentiate $v$ : Differentiate $v = \ln(x)$ . $v' = \frac{1}{x}$ .
Use Product Rule Formula: Apply the product rule: $u*v$ ' = u'v + uv'.
Substitute into Formula: Substitute $u$ , $u'$ , $v$ , and $v'$ into the product rule formula: $(-\frac{5}{x^2})\ln(x) + (\frac{5}{x})(\frac{1}{x})$ .
Simplify Expression: Simplify the expression: $-\frac{5\ln(x)}{x^2} + \frac{5}{x^2}$ .
Combine Like Terms: Combine like terms: $\frac{-5\ln(x) + 5}{x^2}$ .