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Find the difference quotient of \(\displaystyle f(x)=9 x^{2} - x + 2\) .

The difference quotient is \(\displaystyle \frac{f(x+h)-f(x)}{h}\). Evaluating \(\displaystyle f(x+h)=- h - x + 9 \left(h + x\right)^{2} + 2\) and expanding gives \(\displaystyle f(x+h)=9 h^{2} + 18 h x - h + 9 x^{2} - x + 2\) Evaluating the difference quotient gives \(\displaystyle \frac{(9 h^{2} + 18 h x - h + 9 x^{2} - x + 2)-(9 x^{2} - x + 2)}{h}\) Simplifying gives \(\displaystyle \frac{9 h^{2} + 18 h x - h}{h}=9 h + 18 x - 1\)

Download \(\LaTeX\) 

\begin{question}Find the difference quotient of $f(x)=9 x^{2} - x + 2$ . 
    \soln{9cm}{The difference quotient is $\frac{f(x+h)-f(x)}{h}$. Evaluating $f(x+h)=- h - x + 9 \left(h + x\right)^{2} + 2$ and expanding gives $f(x+h)=9 h^{2} + 18 h x - h + 9 x^{2} - x + 2$  Evaluating the difference quotient gives $\frac{(9 h^{2} + 18 h x - h + 9 x^{2} - x + 2)-(9 x^{2} - x + 2)}{h}$ Simplifying gives $\frac{9 h^{2} + 18 h x - h}{h}=9 h + 18 x - 1$ }

\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)=9 x^{2} - x + 2 " src="/equation_images/%20%5Cdisplaystyle%20f%28x%29%3D9%20x%5E%7B2%7D%20-%20x%20%2B%202%20" alt="LaTeX:  \displaystyle f(x)=9 x^{2} - x + 2 " data-equation-content=" \displaystyle f(x)=9 x^{2} - x + 2 " />  . </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 + 9 \left(h + x\right)^{2} + 2 " src="/equation_images/%20%5Cdisplaystyle%20f%28x%2Bh%29%3D-%20h%20-%20x%20%2B%209%20%5Cleft%28h%20%2B%20x%5Cright%29%5E%7B2%7D%20%2B%202%20" alt="LaTeX:  \displaystyle f(x+h)=- h - x + 9 \left(h + x\right)^{2} + 2 " data-equation-content=" \displaystyle f(x+h)=- h - x + 9 \left(h + x\right)^{2} + 2 " />  and expanding gives  <img class="equation_image" title=" \displaystyle f(x+h)=9 h^{2} + 18 h x - h + 9 x^{2} - x + 2 " src="/equation_images/%20%5Cdisplaystyle%20f%28x%2Bh%29%3D9%20h%5E%7B2%7D%20%2B%2018%20h%20x%20-%20h%20%2B%209%20x%5E%7B2%7D%20-%20x%20%2B%202%20" alt="LaTeX:  \displaystyle f(x+h)=9 h^{2} + 18 h x - h + 9 x^{2} - x + 2 " data-equation-content=" \displaystyle f(x+h)=9 h^{2} + 18 h x - h + 9 x^{2} - x + 2 " />   Evaluating the difference quotient gives  <img class="equation_image" title=" \displaystyle \frac{(9 h^{2} + 18 h x - h + 9 x^{2} - x + 2)-(9 x^{2} - x + 2)}{h} " src="/equation_images/%20%5Cdisplaystyle%20%5Cfrac%7B%289%20h%5E%7B2%7D%20%2B%2018%20h%20x%20-%20h%20%2B%209%20x%5E%7B2%7D%20-%20x%20%2B%202%29-%289%20x%5E%7B2%7D%20-%20x%20%2B%202%29%7D%7Bh%7D%20" alt="LaTeX:  \displaystyle \frac{(9 h^{2} + 18 h x - h + 9 x^{2} - x + 2)-(9 x^{2} - x + 2)}{h} " data-equation-content=" \displaystyle \frac{(9 h^{2} + 18 h x - h + 9 x^{2} - x + 2)-(9 x^{2} - x + 2)}{h} " />  Simplifying gives  <img class="equation_image" title=" \displaystyle \frac{9 h^{2} + 18 h x - h}{h}=9 h + 18 x - 1 " src="/equation_images/%20%5Cdisplaystyle%20%5Cfrac%7B9%20h%5E%7B2%7D%20%2B%2018%20h%20x%20-%20h%7D%7Bh%7D%3D9%20h%20%2B%2018%20x%20-%201%20" alt="LaTeX:  \displaystyle \frac{9 h^{2} + 18 h x - h}{h}=9 h + 18 x - 1 " data-equation-content=" \displaystyle \frac{9 h^{2} + 18 h x - h}{h}=9 h + 18 x - 1 " />  </p> </p>