find the simplified difference quotient for the given function.
f(x)=\frac{14}{x^{2}}
find the expression for ( f(x + h) ).
( f(x + h)=square ) (do not simplify.)
( \frac{f(x + h)-f(x)}{h}=square ) (simplify your answer.)

Question

find the simplified difference quotient for the given function.
 f(x)=\frac{14}{x^{2}}
find the expression for ( f(x + h) ).
( f(x + h)=square ) (do not simplify.)
( \frac{f(x + h)-f(x)}{h}=square ) (simplify your answer.)

Accepted Answer

# Explanation:
## Step1: Find f(x + h)
Substitute $x+h$ into $f(x)$: $f(x + h)=\frac{14}{(x + h)^2}$
## Step2: Calculate f(x + h)-f(x)
$$
\begin{align*}
f(x + h)-f(x)&=\frac{14}{(x + h)^2}-\frac{14}{x^2}\
&=\frac{14x^2-14(x + h)^2}{x^2(x + h)^2}
\end{align*}
$$
## Step3: Calculate the difference - quotient
$$
\begin{align*}
\frac{f(x + h)-f(x)}{h}&=\frac{\frac{14x^2-14(x + h)^2}{x^2(x + h)^2}}{h}\
&=\frac{14x^2-14(x^2 + 2xh+h^2)}{hx^2(x + h)^2}\
&=\frac{14x^2-14x^2-28xh - 14h^2}{hx^2(x + h)^2}\
&=\frac{-28xh-14h^2}{hx^2(x + h)^2}\
&=\frac{-14h(2x + h)}{hx^2(x + h)^2}\
&=\frac{-14(2x + h)}{x^2(x + h)^2}
\end{align*}
$$
# Answer:
$f(x + h)=\frac{14}{(x + h)^2}$, $\frac{f(x + h)-f(x)}{h}=\frac{-14(2x + h)}{x^2(x + h)^2}$

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QUESTION IMAGE

Question

Explanation:

Step1: Find f(x + h)

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

Step3: Calculate the difference - quotient

Answer: