Find the difference quotient $$ \frac{f(x+h)-f(x)}{h} $$, where $$ h
eq 0 $$, for the function below. $$ f(x)=\frac{x}{x-9} $$ Simplify your answer as much as possible.

Question

Find the difference quotient $$ \frac{f(x+h)-f(x)}{h} $$, where $$ h 
eq 0 $$, for the function below. $$ f(x)=\frac{x}{x-9} $$ Simplify your answer as much as possible.

UpStudy AI · Accepted Answer

We are given
$$
f(x)=\frac{x}{x-9}
$$
and we want to compute the difference quotient
$$
\frac{f(x+h)-f(x)}{h}
$$
with $$ h\neq 0 $$.

**Step 1. Compute $$ f(x+h) $$:**

$$
f(x+h)=\frac{x+h}{(x+h)-9}=\frac{x+h}{x+h-9}
$$

**Step 2. Write the difference $$ f(x+h)-f(x) $$:**

$$
f(x+h)-f(x)= \frac{x+h}{x+h-9}-\frac{x}{x-9}
$$

**Step 3. Combine the fractions over a common denominator:**

The common denominator is $$(x+h-9)(x-9)$$. Thus,
$$
f(x+h)-f(x)=\frac{(x+h)(x-9)-x(x+h-9)}{(x+h-9)(x-9)}
$$

**Step 4. Expand and simplify the numerator:**

First, expand $$(x+h)(x-9)$$:
$$
(x+h)(x-9)= x^2 - 9x + hx - 9h
$$

Next, expand $$x(x+h-9)$$:
$$
x(x+h-9)= x^2 + xh - 9x
$$

Now, subtract the second expansion from the first:
$$
\begin{aligned}
(x^2 - 9x + hx - 9h) - (x^2 + xh - 9x) &= x^2 - 9x + hx - 9h - x^2 - xh + 9x\[1mm]
&= -9h
\end{aligned}
$$

**Step 5. Write the expression for the difference quotient:**

Substituting the simplified numerator back:
$$
\frac{f(x+h)-f(x)}{h} = \frac{\frac{-9h}{(x+h-9)(x-9)}}{h}
$$

Simplify by dividing by $$h$$ (noting that $$ h\neq 0 $$):
$$
\frac{f(x+h)-f(x)}{h} = \frac{-9h}{h\,(x+h-9)(x-9)} = \frac{-9}{(x+h-9)(x-9)}
$$

Thus, the simplified difference quotient is:
$$
\boxed{\frac{-9}{(x+h-9)(x-9)}}
$$
The difference quotient simplifies to $$ \frac{-9}{(x+h-9)(x-9)} $$.

Find the difference quotient , where , for the function below.

Simplify your answer as much as possible.

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Find the difference quotient , where , for the function below. Simplify your answer as much as possible.