Questions: Find the derivative: f(x)=ln (x /(x-1)) -1 /[x(x+1)] 1 /[x(x-1)] None of these. -1 /x(x-1)

Transcript text: Find the derivative: $f(x)=\ln (x /(x-1))$ $-1 /[x(x+1)]$ $1 /[x(x-1)]$ None of these. $-1 /\{x(x-1)\}$

Solution

Solution Steps

Step 1: Rewrite the function

The given function is: \[ f(x) = \ln \left( \frac{x}{x-1} \right) \] Using the logarithm property $\ln \left( \frac{a}{b} \right) = \ln a - \ln b$, we rewrite the function as: \[ f(x) = \ln x - \ln (x-1) \]

Step 2: Differentiate term by term

Differentiate $f(x)$ with respect to $x$: \[ f'(x) = \frac{d}{dx} [\ln x] - \frac{d}{dx} [\ln (x-1)] \] The derivative of $\ln x$ is $\frac{1}{x}$, and the derivative of $\ln (x-1)$ is $\frac{1}{x-1}$. Thus: \[ f'(x) = \frac{1}{x} - \frac{1}{x-1} \]

Step 3: Simplify the expression

Combine the terms over a common denominator: \[ f'(x) = \frac{(x-1) - x}{x(x-1)} = \frac{-1}{x(x-1)} \] Simplify: \[ f'(x) = -\frac{1}{x(x-1)} \]