Final answer:
To find the derivative of ln(sqrt(x * (1/x - 1))), simplify the expression and then apply the chain rule along with the properties of logarithms and exponents to differentiate the function.
Step-by-step explanation:
The derivative of the natural logarithm of the square root of (x multiplied by 1/x minus 1), which can be written as ln(sqrt(x * (1/x - 1))), can be calculated using the chain rule and properties of logarithms and exponents. First, recall the property of logarithms that allows us to write the logarithm of a square root as one-half the logarithm. Additionally, the negative exponent rule tells us that x-1 equals 1/x. Using the property that ln(xy) = ln(x) + ln(y) and the property that the ln(x/y) = ln(x) - ln(y), we can rewrite the function within the logarithm appropriately, simplifying it before applying the differentiation rules. After simplifying, we apply the chain rule to take the derivative of the natural logarithm, then apply the product or quotient rule as needed to differentiate the inner function.