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Using the difference quotient \(\displaystyle \lim_{h \to 0}\frac{f(x+h)-f(x)}{h}\) gives \(\displaystyle \lim_{h\to 0}\frac{\sqrt{9 h + 9 x - 8}-\sqrt{9 x - 8}}{h}\) Building the fraction using the conjugate gives: \(\displaystyle \frac{- \sqrt{9 x - 8} + \sqrt{9 h + 9 x - 8}}{h}\cdot\left( \frac{\sqrt{9 x - 8} + \sqrt{9 h + 9 x - 8}}{\sqrt{9 x - 8} + \sqrt{9 h + 9 x - 8}} \right)\), expanding and collecting like terms gives \(\displaystyle \lim_{h \to 0}\frac{9 h}{h \left(\sqrt{9 x - 8} + \sqrt{9 h + 9 x - 8}\right)}\) Factoring out \(\displaystyle h\) and reducing gives \(\displaystyle \lim_{h\to 0}\frac{1}{\sqrt{9 x - 8} + \sqrt{9 h + 9 x - 8}}\). Evaluating the limit gives \(\displaystyle f'(x)=\frac{9}{2 \sqrt{9 x - 8}}\)

Download \(\LaTeX\)

\begin{question}Use the limit definition of the derivative to find $f'(x)$ if $f(x)= \sqrt{9 x - 8}$. 
    \soln{9cm}{Using the difference quotient $\lim_{h \to 0}\frac{f(x+h)-f(x)}{h}$ gives $\lim_{h\to 0}\frac{\sqrt{9 h + 9 x - 8}-\sqrt{9 x - 8}}{h}$ Building the fraction using the conjugate gives: $\frac{- \sqrt{9 x - 8} + \sqrt{9 h + 9 x - 8}}{h}\cdot\left( \frac{\sqrt{9 x - 8} + \sqrt{9 h + 9 x - 8}}{\sqrt{9 x - 8} + \sqrt{9 h + 9 x - 8}} \right)$, expanding and collecting like terms gives $\lim_{h \to 0}\frac{9 h}{h \left(\sqrt{9 x - 8} + \sqrt{9 h + 9 x - 8}\right)}$ Factoring out $h$ and reducing gives $\lim_{h\to 0}\frac{1}{\sqrt{9 x - 8} + \sqrt{9 h + 9 x - 8}}$. Evaluating the limit gives $f'(x)=\frac{9}{2 \sqrt{9 x - 8}}$}

\end{question}

Download Question and Solution Environment\(\LaTeX\)

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\begin{document}\begin{question}(10pts) The question goes here!
    \soln{9cm}{The solution goes here.}

\end{question}\end{document}

HTML for Canvas

<p> <p>Use the limit definition of the derivative to find  <img class="equation_image" title=" \displaystyle f'(x) " src="/equation_images/%20%5Cdisplaystyle%20f%27%28x%29%20" alt="LaTeX:  \displaystyle f'(x) " data-equation-content=" \displaystyle f'(x) " />  if  <img class="equation_image" title=" \displaystyle f(x)= \sqrt{9 x - 8} " src="/equation_images/%20%5Cdisplaystyle%20f%28x%29%3D%20%5Csqrt%7B9%20x%20-%208%7D%20" alt="LaTeX:  \displaystyle f(x)= \sqrt{9 x - 8} " data-equation-content=" \displaystyle f(x)= \sqrt{9 x - 8} " /> . </p> </p>

HTML for Canvas

<p> <p>Using the difference quotient  <img class="equation_image" title=" \displaystyle \lim_{h \to 0}\frac{f(x+h)-f(x)}{h} " src="/equation_images/%20%5Cdisplaystyle%20%5Clim_%7Bh%20%5Cto%200%7D%5Cfrac%7Bf%28x%2Bh%29-f%28x%29%7D%7Bh%7D%20" alt="LaTeX:  \displaystyle \lim_{h \to 0}\frac{f(x+h)-f(x)}{h} " data-equation-content=" \displaystyle \lim_{h \to 0}\frac{f(x+h)-f(x)}{h} " />  gives  <img class="equation_image" title=" \displaystyle \lim_{h\to 0}\frac{\sqrt{9 h + 9 x - 8}-\sqrt{9 x - 8}}{h} " src="/equation_images/%20%5Cdisplaystyle%20%5Clim_%7Bh%5Cto%200%7D%5Cfrac%7B%5Csqrt%7B9%20h%20%2B%209%20x%20-%208%7D-%5Csqrt%7B9%20x%20-%208%7D%7D%7Bh%7D%20" alt="LaTeX:  \displaystyle \lim_{h\to 0}\frac{\sqrt{9 h + 9 x - 8}-\sqrt{9 x - 8}}{h} " data-equation-content=" \displaystyle \lim_{h\to 0}\frac{\sqrt{9 h + 9 x - 8}-\sqrt{9 x - 8}}{h} " />  Building the fraction using the conjugate gives:  <img class="equation_image" title=" \displaystyle \frac{- \sqrt{9 x - 8} + \sqrt{9 h + 9 x - 8}}{h}\cdot\left( \frac{\sqrt{9 x - 8} + \sqrt{9 h + 9 x - 8}}{\sqrt{9 x - 8} + \sqrt{9 h + 9 x - 8}} \right) " src="/equation_images/%20%5Cdisplaystyle%20%5Cfrac%7B-%20%5Csqrt%7B9%20x%20-%208%7D%20%2B%20%5Csqrt%7B9%20h%20%2B%209%20x%20-%208%7D%7D%7Bh%7D%5Ccdot%5Cleft%28%20%5Cfrac%7B%5Csqrt%7B9%20x%20-%208%7D%20%2B%20%5Csqrt%7B9%20h%20%2B%209%20x%20-%208%7D%7D%7B%5Csqrt%7B9%20x%20-%208%7D%20%2B%20%5Csqrt%7B9%20h%20%2B%209%20x%20-%208%7D%7D%20%5Cright%29%20" alt="LaTeX:  \displaystyle \frac{- \sqrt{9 x - 8} + \sqrt{9 h + 9 x - 8}}{h}\cdot\left( \frac{\sqrt{9 x - 8} + \sqrt{9 h + 9 x - 8}}{\sqrt{9 x - 8} + \sqrt{9 h + 9 x - 8}} \right) " data-equation-content=" \displaystyle \frac{- \sqrt{9 x - 8} + \sqrt{9 h + 9 x - 8}}{h}\cdot\left( \frac{\sqrt{9 x - 8} + \sqrt{9 h + 9 x - 8}}{\sqrt{9 x - 8} + \sqrt{9 h + 9 x - 8}} \right) " /> , expanding and collecting like terms gives  <img class="equation_image" title=" \displaystyle \lim_{h \to 0}\frac{9 h}{h \left(\sqrt{9 x - 8} + \sqrt{9 h + 9 x - 8}\right)} " src="/equation_images/%20%5Cdisplaystyle%20%5Clim_%7Bh%20%5Cto%200%7D%5Cfrac%7B9%20h%7D%7Bh%20%5Cleft%28%5Csqrt%7B9%20x%20-%208%7D%20%2B%20%5Csqrt%7B9%20h%20%2B%209%20x%20-%208%7D%5Cright%29%7D%20" alt="LaTeX:  \displaystyle \lim_{h \to 0}\frac{9 h}{h \left(\sqrt{9 x - 8} + \sqrt{9 h + 9 x - 8}\right)} " data-equation-content=" \displaystyle \lim_{h \to 0}\frac{9 h}{h \left(\sqrt{9 x - 8} + \sqrt{9 h + 9 x - 8}\right)} " />  Factoring out  <img class="equation_image" title=" \displaystyle h " src="/equation_images/%20%5Cdisplaystyle%20h%20" alt="LaTeX:  \displaystyle h " data-equation-content=" \displaystyle h " />  and reducing gives  <img class="equation_image" title=" \displaystyle \lim_{h\to 0}\frac{1}{\sqrt{9 x - 8} + \sqrt{9 h + 9 x - 8}} " src="/equation_images/%20%5Cdisplaystyle%20%5Clim_%7Bh%5Cto%200%7D%5Cfrac%7B1%7D%7B%5Csqrt%7B9%20x%20-%208%7D%20%2B%20%5Csqrt%7B9%20h%20%2B%209%20x%20-%208%7D%7D%20" alt="LaTeX:  \displaystyle \lim_{h\to 0}\frac{1}{\sqrt{9 x - 8} + \sqrt{9 h + 9 x - 8}} " data-equation-content=" \displaystyle \lim_{h\to 0}\frac{1}{\sqrt{9 x - 8} + \sqrt{9 h + 9 x - 8}} " /> . Evaluating the limit gives  <img class="equation_image" title=" \displaystyle f'(x)=\frac{9}{2 \sqrt{9 x - 8}} " src="/equation_images/%20%5Cdisplaystyle%20f%27%28x%29%3D%5Cfrac%7B9%7D%7B2%20%5Csqrt%7B9%20x%20-%208%7D%7D%20" alt="LaTeX:  \displaystyle f'(x)=\frac{9}{2 \sqrt{9 x - 8}} " data-equation-content=" \displaystyle f'(x)=\frac{9}{2 \sqrt{9 x - 8}} " /> </p> </p>