Find the difference quotient \( \frac{f(x+h)-f(x)}{h} \), where \( h \neq 0 \), for the function below. \( f(x)=\frac{x}{x+1} \) Simplify your answer as much as possible. \( \frac{f(x+h)-f(x)}{h}= \) Ø.

We start with the function \[ f(x) = \frac{x}{x+1}, \] so its value at \( x+h \) is \[ f(x+h) = \frac{x+h}{x+h+1}. \] The difference quotient is defined as \[ \frac{f(x+h)-f(x)}{h} = \frac{\frac{x+h}{x+h+1} - \frac{x}{x+1}}{h}. \] **Step 1. Combine the fractions in the numerator** Write the difference as a single fraction: \[ \frac{x+h}{x+h+1} - \frac{x}{x+1} = \frac{(x+h)(x+1) - x(x+h+1)}{(x+h+1)(x+1)}. \] **Step 2. Simplify the numerator** Expand each term in the numerator: \[ \begin{aligned} (x+h)(x+1) &= x(x+1) + h(x+1) = x^2 + x + hx + h, \\ x(x+h+1) &= x^2 + xh + x. \end{aligned} \] Subtract the second expression from the first: \[ (x^2 + x + hx + h) - (x^2 + xh + x) = h. \] Thus, the numerator simplifies to: \[ \frac{(x+h)(x+1) - x(x+h+1)}{(x+h+1)(x+1)} = \frac{h}{(x+h+1)(x+1)}. \] **Step 3. Divide by \( h \)** Substitute back into the difference quotient: \[ \frac{f(x+h)-f(x)}{h} = \frac{\frac{h}{(x+h+1)(x+1)}}{h} = \frac{1}{(x+h+1)(x+1)}. \] Thus, the simplified difference quotient is: \[ \frac{f(x+h)-f(x)}{h} = \frac{1}{(x+1)(x+h+1)}. \]