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

Identify Function: We need to find the derivative of the function (ln(x))/(sqrt(x)) with respect to x. This requires the use of the quotient rule for derivatives, which states that if you have a function h(x) = f(x)/g(x), then h'(x) = (f'(x)g(x) - f(x)g'(x))/(g(x))^2. Here, f(x) = ln(x) and g(x) = sqrt(x) or x^(1/2).Derivative of ln(x): First, we find the derivative of f(x) = ln(x). The derivative of ln(x) with respect to x is 1/x.Derivative of sqrt(x): Next, we find the derivative of g(x) = sqrt(x) or x^(1/2). Using the power rule, the derivative of x^(1/2) is (1/2)x^(-1/2) or 1/(2sqrt(x)).Apply Quotient Rule: Now we apply the quotient rule. We have f'(x) = 1/x and g'(x) = 1/(2sqrt(x)). Plugging these into the quotient rule formula, we get: h'(x) = ((1/x) * (sqrt(x)) - (ln(x)) * (1/(2sqrt(x)))) / (sqrt(x))^2 Simplify the numerator: h'(x) = (sqrt(x)/x - ln(x)/(2sqrt(x))) / (x)Simplify Numerator: Simplify the expression further by finding a common denominator for the terms in the numerator: h'(x) = ((2 - ln(x))/(2sqrt(x))) / x Now, divide each term in the numerator by x: h'(x) = (2/(2x^(3/2))) - (ln(x)/(2x^(3/2)))Find Common Denominator: Simplify the terms: h'(x) = (1/x^(3/2)) - (ln(x)/(2x^(3/2))) Combine the terms over the common denominator: h'(x) = (2 - ln(x))/(2x^(3/2))

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

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Algebra 2

Simplify variable expressions using properties

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

Identify Function: We need to find the derivative of the function $\frac{\ln(x)}{\sqrt{x}}$ with respect to $x$ . This requires the use of the quotient rule for derivatives, which states that if you have a function $h(x) = \frac{f(x)}{g(x)}$ , then $h'(x) = \frac{f'(x)g(x) - f(x)g'(x)}{(g(x))^2}$ . Here, $f(x) = \ln(x)$ and $g(x) = \sqrt{x}$ or $x^{\frac{1}{2}}$ .
Derivative of ln(x): First, we find the derivative of $f(x) = \ln(x)$ . The derivative of $\ln(x)$ with respect to $x$ is $\frac{1}{x}$ .
Derivative of $\sqrt{x}$ : Next, we find the derivative of $g(x) = \sqrt{x}$ or $x^{(1/2)}$ . Using the power rule, the derivative of $x^{(1/2)}$ is $(1/2)x^{(-1/2)}$ or $\frac{1}{2\sqrt{x}}$ .
Apply Quotient Rule: Now we apply the quotient rule. We have $f'(x) = \frac{1}{x}$ and $g'(x) = \frac{1}{2\sqrt{x}}$ . Plugging these into the quotient rule formula, we get: $\newline$ $h'(x) = \frac{(\frac{1}{x}) * (\sqrt{x}) - (\ln(x)) * (\frac{1}{2\sqrt{x}})}{(\sqrt{x})^2}$ $\newline$ Simplify the numerator: $\newline$ $h'(x) = \frac{\sqrt{x}/x - \ln(x)/(2\sqrt{x})}{x}$
Simplify Numerator: Simplify the expression further by finding a common denominator for the terms in the numerator: $\newline$ $h'(x) = \frac{2 - \ln(x)}{2\sqrt{x}} / x$ $\newline$ Now, divide each term in the numerator by $x$ : $\newline$ $h'(x) = \frac{2}{2x^{\frac{3}{2}}} - \frac{\ln(x)}{2x^{\frac{3}{2}}}$
Find Common Denominator: Simplify the terms: $\newline$ $h'(x) = \frac{1}{x^{\frac{3}{2}}} - \frac{\ln(x)}{2x^{\frac{3}{2}}}$ $\newline$ Combine the terms over the common denominator: $\newline$ $h'(x) = \frac{2 - \ln(x)}{2x^{\frac{3}{2}}}$