To find the derivative of the function:

$f(x) = \frac{\ln(x - 1) - 1}{(\ln(x - 1))^2}$

we will use the quotient rule. The quotient rule states that for a function of the form $f(x) = \frac{u(x)}{v(x)}$, the derivative $f'(x)$ is given by:

$f'(x) = \frac{u'(x)v(x) - u(x)v'(x)}{(v(x))^2}.$

Step 1: Define $u(x)$ and $v(x)$

• $u(x) = \ln(x - 1) - 1$
• $v(x) = (\ln(x - 1))^2$

Step 2: Compute $u'(x)$ and $v'(x)$

Derivative of $u(x)$

$u(x) = \ln(x - 1) - 1$

Using the chain rule:

$u'(x) = \frac{1}{x - 1}.$

Derivative of $v(x)$

$v(x) = (\ln(x - 1))^2$

Using the chain rule and the power rule:

$v'(x) = 2 \ln(x - 1) \cdot \frac{1}{x - 1} = \frac{2 \ln(x - 1)}{x - 1}.$

Step 3: Apply the Quotient Rule

Substituting $u(x)$, $v(x)$, $u'(x)$, and $v'(x)$ into the quotient rule:

$f'(x) = \frac{u'(x)v(x) - u(x)v'(x)}{(v(x))^2}.$

Plugging in the expressions:

$f'(x) = \frac{\left(\frac{1}{x - 1}\right)(\ln(x - 1))^2 - (\ln(x - 1) - 1)\left(\frac{2 \ln(x - 1)}{x - 1}\right)}{(\ln(x - 1)^2)^2}.$

Step 4: Simplify the Numerator

Factor out $\frac{1}{x - 1}$ from both terms in the numerator:

$f'(x) = \frac{\frac{1}{x - 1} \left[ (\ln(x - 1))^2 - 2\ln(x - 1)(\ln(x - 1) - 1) \right]}{(\ln(x - 1))^4}.$

Simplify inside the brackets:

$(\ln(x - 1))^2 - 2\ln(x - 1)(\ln(x - 1) - 1) = (\ln(x - 1))^2 - 2(\ln(x - 1))^2 + 2\ln(x - 1).$

Combine like terms:

$= -(\ln(x - 1))^2 + 2\ln(x - 1).$

Thus, the expression for $f'(x)$ becomes:

$f'(x) = \frac{\frac{1}{x - 1} \left[ -(\ln(x - 1))^2 + 2\ln(x - 1) \right]}{(\ln(x - 1))^4}.$

Step 5: Final Expression

Combining the terms:

$f'(x) = \frac{-(\ln(x - 1))^2 + 2\ln(x - 1)}{(x - 1)(\ln(x - 1))^4}.$

This is the derivative of the function $f(x)$.

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Do you need a more detailed explanation or simplification?

Related Questions:

1. What is the derivative of $g(x) = \frac{\ln(x)}{(\ln(x))^2}$?
2. How do you apply the quotient rule to $f(x) = \frac{e^x}{x^2}$?
3. What is the second derivative of $f(x) = \frac{\ln(x - 1) - 1}{(\ln(x - 1))^2}$?
4. How do you simplify expressions involving logarithmic derivatives?
5. What are the critical points of $f(x) = \frac{\ln(x - 1) - 1}{(\ln(x - 1))^2}$?

Tip:

When dealing with derivatives of logarithmic functions, always double-check if the chain rule applies.