Let h(x)=(ln(x))/(x^(2)). h^(')(x)=

Identify Function: We need to find the derivative of the function h(x) = (ln(x))/(x^2). To do this, we will use the quotient rule for derivatives, which states that if we have a function that is the quotient of two functions, u(x)/v(x), then its derivative is given by (v(x)u'(x) - u(x)v'(x))/(v(x))^2. Here, u(x) = ln(x) and v(x) = x^2.Derivative of ln(x): First, we find the derivative of u(x) = ln(x). The derivative of ln(x) with respect to x is 1/x. u'(x) = 1/xDerivative of x^2: Next, we find the derivative of v(x) = x^2. The derivative of x^2 with respect to x is 2x. v'(x) = 2xApply Quotient Rule: Now we apply the quotient rule. We have: h'(x) = (v(x)u'(x) - u(x)v'(x)) / (v(x))^2 Substituting u(x), u'(x), v(x), and v'(x) into the formula, we get: h'(x) = (x^2 * (1/x) - ln(x) * 2x) / (x^2)^2Simplify Expression: Simplify the expression by performing the multiplication and division: h'(x) = (x - 2x * ln(x)) / x^4Factor Out x: We can further simplify by factoring out an x from the numerator: h'(x) = x(1 - 2ln(x)) / x^4Cancel x: Finally, we can cancel one x from the numerator and denominator: h'(x) = (1 - 2ln(x)) / x^3 This is the simplified form of the derivative of h(x).

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Algebra 1

Write variable expressions for arithmetic sequences

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Q. Let

h(x)=\frac{\ln (x)}{x^{2}}

\newline

h^{\prime}(x)=

Identify Function: We need to find the derivative of the function $h(x) = \frac{\ln(x)}{x^2}$ . To do this, we will use the quotient rule for derivatives, which states that if we have a function that is the quotient of two functions, $\frac{u(x)}{v(x)}$ , then its derivative is given by $\frac{v(x)u'(x) - u(x)v'(x)}{(v(x))^2}$ . Here, $u(x) = \ln(x)$ and $v(x) = x^2$ .
Derivative of ln(x): First, we find the derivative of $u(x) = \ln(x)$ . The derivative of $\ln(x)$ with respect to $x$ is $\frac{1}{x}$ . $\newline$ $u'(x) = \frac{1}{x}$
Derivative of $x^2$ : Next, we find the derivative of $v(x) = x^2$ . The derivative of $x^2$ with respect to $x$ is $2x$ . $\newline$ $v'(x) = 2x$
Apply Quotient Rule: Now we apply the quotient rule. We have: $\newline$ $h'(x) = \frac{v(x)u'(x) - u(x)v'(x)}{(v(x))^2}$ $\newline$ Substituting $u(x)$ , $u'(x)$ , $v(x)$ , and $v'(x)$ into the formula, we get: $\newline$ $h'(x) = \frac{x^2 \cdot (\frac{1}{x}) - \ln(x) \cdot 2x}{(x^2)^2}$
Simplify Expression: Simplify the expression by performing the multiplication and division: $h'(x) = \frac{x - 2x \cdot \ln(x)}{x^4}$
Factor Out x: We can further simplify by factoring out an $x$ from the numerator: $h'(x) = \frac{x(1 - 2\ln(x))}{x^4}$
Cancel $x$ : Finally, we can cancel one $x$ from the numerator and denominator: $\newline$ $h'(x) = \frac{1 - 2\ln(x)}{x^3}$ $\newline$ This is the simplified form of the derivative of $h(x)$ .