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Find the difference quotient of \(\displaystyle f(x)=10 - x\) .

The difference quotient is \(\displaystyle \frac{f(x+h)-f(x)}{h}\). Evaluating \(\displaystyle f(x+h)=- h - x + 10\) and expanding gives \(\displaystyle f(x+h)=- h - x + 10\) Evaluating the difference quotient gives \(\displaystyle \frac{(- h - x + 10)-(10 - x)}{h}\) Simplifying gives \(\displaystyle \frac{- h}{h}=-1\)

Download \(\LaTeX\) 

\begin{question}Find the difference quotient of $f(x)=10 - x$ . 
    \soln{9cm}{The difference quotient is $\frac{f(x+h)-f(x)}{h}$. Evaluating $f(x+h)=- h - x + 10$ and expanding gives $f(x+h)=- h - x + 10$  Evaluating the difference quotient gives $\frac{(- h - x + 10)-(10 - x)}{h}$ Simplifying gives $\frac{- h}{h}=-1$ }

\end{question}

Download Question and Solution Environment\(\LaTeX\)
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HTML for Canvas 
<p> <p>Find the difference quotient of  <img class="equation_image" title=" \displaystyle f(x)=10 - x " src="/equation_images/%20%5Cdisplaystyle%20f%28x%29%3D10%20-%20x%20" alt="LaTeX:  \displaystyle f(x)=10 - x " data-equation-content=" \displaystyle f(x)=10 - x " />  . </p> </p>
HTML for Canvas
<p> <p>The difference quotient is  <img class="equation_image" title=" \displaystyle \frac{f(x+h)-f(x)}{h} " src="/equation_images/%20%5Cdisplaystyle%20%5Cfrac%7Bf%28x%2Bh%29-f%28x%29%7D%7Bh%7D%20" alt="LaTeX:  \displaystyle \frac{f(x+h)-f(x)}{h} " data-equation-content=" \displaystyle \frac{f(x+h)-f(x)}{h} " /> . Evaluating  <img class="equation_image" title=" \displaystyle f(x+h)=- h - x + 10 " src="/equation_images/%20%5Cdisplaystyle%20f%28x%2Bh%29%3D-%20h%20-%20x%20%2B%2010%20" alt="LaTeX:  \displaystyle f(x+h)=- h - x + 10 " data-equation-content=" \displaystyle f(x+h)=- h - x + 10 " />  and expanding gives  <img class="equation_image" title=" \displaystyle f(x+h)=- h - x + 10 " src="/equation_images/%20%5Cdisplaystyle%20f%28x%2Bh%29%3D-%20h%20-%20x%20%2B%2010%20" alt="LaTeX:  \displaystyle f(x+h)=- h - x + 10 " data-equation-content=" \displaystyle f(x+h)=- h - x + 10 " />   Evaluating the difference quotient gives  <img class="equation_image" title=" \displaystyle \frac{(- h - x + 10)-(10 - x)}{h} " src="/equation_images/%20%5Cdisplaystyle%20%5Cfrac%7B%28-%20h%20-%20x%20%2B%2010%29-%2810%20-%20x%29%7D%7Bh%7D%20" alt="LaTeX:  \displaystyle \frac{(- h - x + 10)-(10 - x)}{h} " data-equation-content=" \displaystyle \frac{(- h - x + 10)-(10 - x)}{h} " />  Simplifying gives  <img class="equation_image" title=" \displaystyle \frac{- h}{h}=-1 " src="/equation_images/%20%5Cdisplaystyle%20%5Cfrac%7B-%20h%7D%7Bh%7D%3D-1%20" alt="LaTeX:  \displaystyle \frac{- h}{h}=-1 " data-equation-content=" \displaystyle \frac{- h}{h}=-1 " />  </p> </p>