To find the difference quotient [tex]\(\frac{f(x+h)-f(x)}{h}\)[/tex] for the function [tex]\(f(x) = \frac{x}{x-1}\)[/tex], we proceed as follows:
1. 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]
2. Form the difference quotient:
[tex]\[
\frac{f(x + h) - f(x)}{h} = \frac{\frac{x + h}{x + h - 1} - \frac{x}{x - 1}}{h}
\][/tex]
3. Find a common denominator for the fractions in the numerator:
[tex]\[
\frac{x + h}{x + h - 1} = \frac{(x + h)(x - 1)}{(x + h - 1)(x - 1)}
\][/tex]
[tex]\[
\frac{x}{x - 1} = \frac{x(x + h - 1)}{(x + h - 1)(x - 1)}
\][/tex]
4. Combine the fractions:
[tex]\[
\frac{f(x + h) - f(x)}{h} = \frac{\frac{(x + h)(x - 1) - x(x + h - 1)}{(x + h - 1)(x - 1)}}{h}
\][/tex]
5. Simplify the numerator:
[tex]\[
(x + h)(x - 1) - x(x + h - 1) = (x^2 - x + hx - h) - (x^2 + hx - x) = -x + hx - h + x = h(x - 1)
\][/tex]
So, the expression simplifies to:
[tex]\[
\frac{h(x - 1)}{h(x + h - 1)(x - 1)}
\][/tex]
6. Cancel the [tex]\(h\)[/tex] in the numerator and denominator:
[tex]\[
\frac{h(x - 1)}{h(x + h - 1)(x - 1)} = \frac{(x - 1)}{(x + h - 1)(x - 1)} = \frac{1}{x + h - 1}
\][/tex]
7. Final simplified difference quotient:
[tex]\[
-\frac{1}{hx - h + x^2 - 2x + 1}
\][/tex]
Thus, the difference quotient [tex]\(\frac{f(x+h)-f(x)}{h}\)[/tex] for the function [tex]\(f(x) = \frac{x}{x-1}\)[/tex] simplifies to:
[tex]\[
\frac{(f(x+h) - f(x))}{h} = -\frac{1}{hx - h + x^2 - 2x + 1}
\][/tex]
