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.)

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:

\(f(x + h)=\frac{14}{(x + h)^2}\), \(\frac{f(x + h)-f(x)}{h}=\frac{-14(2x + h)}{x^2(x + h)^2}\)