find the difference quotient of f; that is find $ \frac { f ( x + h ) - f ( x ) } { h } $, $ h \
eq 0 $, for the function $ f ( x ) = \frac { 5 x } { x + 4 } $.
the difference quotient of $ f ( x ) = \frac { 5 x } { x + 4 } $ is $ \square $.
(simplify your answer.)

Question

Accepted Answer

# Explanation:
## Step1: Find $f(x + h)$
Substitute $x+h$ into $f(x)=\frac{5x}{x + 4}$, we get $f(x + h)=\frac{5(x + h)}{(x + h)+4}=\frac{5x+5h}{x + h+4}$.
## Step2: Calculate $f(x + h)-f(x)$
$$
\begin{align*}
f(x + h)-f(x)&=\frac{5x + 5h}{x + h+4}-\frac{5x}{x + 4}\
&=\frac{(5x + 5h)(x + 4)-5x(x + h + 4)}{(x + h+4)(x + 4)}\
&=\frac{5x^{2}+20x+5hx + 20h-(5x^{2}+5hx+20x)}{(x + h+4)(x + 4)}\
&=\frac{5x^{2}+20x+5hx + 20h - 5x^{2}-5hx-20x}{(x + h+4)(x + 4)}\
&=\frac{20h}{(x + h+4)(x + 4)}
\end{align*}
$$
## Step3: Find $\frac{f(x + h)-f(x)}{h}$
$$
\begin{align*}
\frac{f(x + h)-f(x)}{h}&=\frac{\frac{20h}{(x + h+4)(x + 4)}}{h}\
&=\frac{20h}{(x + h+4)(x + 4)}\times\frac{1}{h}\
&=\frac{20}{(x + h+4)(x + 4)}
\end{align*}
$$

# Answer:
$\frac{20}{(x + h+4)(x + 4)}$

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QUESTION IMAGE

Question

Explanation:

Step1: Find \(f(x + h)\)

Step2: Calculate \(f(x + h)-f(x)\)

Step3: Find \(\frac{f(x + h)-f(x)}{h}\)

Answer: