To find the difference quotient for the function [tex]\( f(x) = \frac{x}{x - 1} \)[/tex], we follow these steps:
1. Define [tex]\( f(x) \)[/tex]:
[tex]\[
f(x) = \frac{x}{x - 1}
\][/tex]
2. Calculate [tex]\( f(x + h) \)[/tex]:
Substitute [tex]\( x + h \)[/tex] into the function:
[tex]\[
f(x + h) = \frac{x + h}{(x + h) - 1} = \frac{x + h}{x + h - 1}
\][/tex]
3. Form the difference [tex]\( f(x + h) - f(x) \)[/tex]:
[tex]\[
f(x + h) - f(x) = \frac{x + h}{x + h - 1} - \frac{x}{x - 1}
\][/tex]
4. Find a common denominator to combine the fractions:
[tex]\[
\frac{x + h}{x + h - 1} - \frac{x}{x - 1} = \frac{(x + h)(x - 1) - x(x + h - 1)}{(x + h - 1)(x - 1)}
\][/tex]
5. Simplify the numerator:
Expand the terms in the numerator:
[tex]\[
(x + h)(x - 1) = x^2 - x + hx - h
\][/tex]
[tex]\[
x(x + h - 1) = x^2 + hx - x
\][/tex]
Therefore:
[tex]\[
(x^2 - x + hx - h) - (x^2 + hx - x) = -h
\][/tex]
6. Combine the simplified numerator and denominator:
[tex]\[
\frac{f(x + h) - f(x)}{h} = \frac{\frac{-h}{(x + h - 1)(x - 1)}}{h} = \frac{-h}{h(x + h - 1)(x - 1)}
\][/tex]
7. Cancel out [tex]\( h \)[/tex] in the numerator and denominator (since [tex]\( h \neq 0 \)[/tex]):
[tex]\[
\frac{-h}{h(x + h - 1)(x - 1)} = \frac{-1}{(x + h - 1)(x - 1)}
\][/tex]
Thus, the simplified difference quotient is:
[tex]\[
\frac{f(x+h)-f(x)}{h} = \frac{-1}{(x + h - 1)(x - 1)}
\][/tex]
