Find the derivative of r(x). r(x) = ln(1/x) r′(x) = ______

Recognize logarithm properties: Recognize that r(x) = ln(1/x) can be rewritten using the properties of logarithms as r(x) = -ln(x).Find derivative of r(x): Now that we have r(x) = -ln(x), we can find the derivative of r(x) with respect to x. The derivative of ln(x) with respect to x is 1/x.Apply chain rule: Since r(x) = -ln(x), the derivative r′(x) is the derivative of -ln(x), which is -1 times the derivative of ln(x). Therefore, r′(x) = -1 * (1/x).Simplify the expression: Simplify the expression for r′(x) to get the final answer. r′(x) = -1/x.

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Find the derivative of $r(x)$ . $\newline$ $r(x) = \ln\left(\frac{1}{x}\right)$ $\newline$ $r'(x) =$ ______

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Calculus

Find derivatives using the quotient rule I

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Q. Find the derivative of

r(x)

\newline

r(x) = \ln\left(\frac{1}{x}\right)

\newline

r'(x) =

______

Recognize logarithm properties: Recognize that $r(x) = \ln(\frac{1}{x})$ can be rewritten using the properties of logarithms as $r(x) = -\ln(x)$ .
Find derivative of $r(x)$ : Now that we have $r(x) = -\ln(x)$ , we can find the derivative of $r(x)$ with respect to $x$ . The derivative of $\ln(x)$ with respect to $x$ is $\frac{1}{x}$ .
Apply chain rule: Since $r(x) = -\ln(x)$ , the derivative $r'(x)$ is the derivative of $-\ln(x)$ , which is $-1$ times the derivative of $\ln(x)$ . Therefore, $r'(x) = -1 \times (1/x)$ .
Simplify the expression: Simplify the expression for $r'(x)$ to get the final answer. $r'(x) = -\frac{1}{x}$ .