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Identifying \(\displaystyle f=- 7 x - 9\) and \(\displaystyle g=\left(3 - 4 x\right) \log{\left(x \right)}\) and using the product rule with \(\displaystyle f=- 7 x - 9 \implies f'=-7\). This leaves g as \(\displaystyle g = \left(3 - 4 x\right) \log{\left(x \right)}\) which also requires the product rule. Pushing down in the new product rule \(\displaystyle f=\log{\left(x \right)} \implies f'=\frac{1}{x}\) and \(\displaystyle g=3 - 4 x \implies g'=-4\). Popping up a level gives \(\displaystyle g'=(3 - 4 x)(\frac{1}{x})+(\log{\left(x \right)})(-4)\)Popping up again (Back to the original problem) gives \(\displaystyle f'=(- 7 x - 9)(- 4 \log{\left(x \right)} + \frac{3 - 4 x}{x})+(\left(3 - 4 x\right) \log{\left(x \right)})(-7)=\left(28 x - 21\right) \log{\left(x \right)} + \left(28 x + 36\right) \log{\left(x \right)} + \frac{\left(3 - 4 x\right) \left(- 7 x - 9\right)}{x}\)

Download \(\LaTeX\)

\begin{question}Find the derivative of $y = (- 7 x - 9)(\log{\left(x \right)})(3 - 4 x)$.
    \soln{9cm}{Identifying $f=- 7 x - 9$ and $g=\left(3 - 4 x\right) \log{\left(x \right)}$ and using the product rule with $f=- 7 x - 9 \implies f'=-7$. This leaves g as $g = \left(3 - 4 x\right) \log{\left(x \right)}$ which also requires the product rule. Pushing down in the new product rule $f=\log{\left(x \right)} \implies f'=\frac{1}{x}$ and $g=3 - 4 x \implies g'=-4$. Popping up a level gives $g'=(3 - 4 x)(\frac{1}{x})+(\log{\left(x \right)})(-4)$Popping up again (Back to the original problem) gives $f'=(- 7 x - 9)(- 4 \log{\left(x \right)} + \frac{3 - 4 x}{x})+(\left(3 - 4 x\right) \log{\left(x \right)})(-7)=\left(28 x - 21\right) \log{\left(x \right)} + \left(28 x + 36\right) \log{\left(x \right)} + \frac{\left(3 - 4 x\right) \left(- 7 x - 9\right)}{x}$}

\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.}

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HTML for Canvas

<p> <p>Find the derivative of  <img class="equation_image" title=" \displaystyle y = (- 7 x - 9)(\log{\left(x \right)})(3 - 4 x) " src="/equation_images/%20%5Cdisplaystyle%20y%20%3D%20%28-%207%20x%20-%209%29%28%5Clog%7B%5Cleft%28x%20%5Cright%29%7D%29%283%20-%204%20x%29%20" alt="LaTeX:  \displaystyle y = (- 7 x - 9)(\log{\left(x \right)})(3 - 4 x) " data-equation-content=" \displaystyle y = (- 7 x - 9)(\log{\left(x \right)})(3 - 4 x) " /> .</p> </p>

HTML for Canvas

<p> <p>Identifying  <img class="equation_image" title=" \displaystyle f=- 7 x - 9 " src="/equation_images/%20%5Cdisplaystyle%20f%3D-%207%20x%20-%209%20" alt="LaTeX:  \displaystyle f=- 7 x - 9 " data-equation-content=" \displaystyle f=- 7 x - 9 " />  and  <img class="equation_image" title=" \displaystyle g=\left(3 - 4 x\right) \log{\left(x \right)} " src="/equation_images/%20%5Cdisplaystyle%20g%3D%5Cleft%283%20-%204%20x%5Cright%29%20%5Clog%7B%5Cleft%28x%20%5Cright%29%7D%20" alt="LaTeX:  \displaystyle g=\left(3 - 4 x\right) \log{\left(x \right)} " data-equation-content=" \displaystyle g=\left(3 - 4 x\right) \log{\left(x \right)} " />  and using the product rule with  <img class="equation_image" title=" \displaystyle f=- 7 x - 9 \implies f'=-7 " src="/equation_images/%20%5Cdisplaystyle%20f%3D-%207%20x%20-%209%20%5Cimplies%20f%27%3D-7%20" alt="LaTeX:  \displaystyle f=- 7 x - 9 \implies f'=-7 " data-equation-content=" \displaystyle f=- 7 x - 9 \implies f'=-7 " /> . This leaves g as  <img class="equation_image" title=" \displaystyle g = \left(3 - 4 x\right) \log{\left(x \right)} " src="/equation_images/%20%5Cdisplaystyle%20g%20%3D%20%5Cleft%283%20-%204%20x%5Cright%29%20%5Clog%7B%5Cleft%28x%20%5Cright%29%7D%20" alt="LaTeX:  \displaystyle g = \left(3 - 4 x\right) \log{\left(x \right)} " data-equation-content=" \displaystyle g = \left(3 - 4 x\right) \log{\left(x \right)} " />  which also requires the product rule. Pushing down in the new product rule  <img class="equation_image" title=" \displaystyle f=\log{\left(x \right)} \implies f'=\frac{1}{x} " src="/equation_images/%20%5Cdisplaystyle%20f%3D%5Clog%7B%5Cleft%28x%20%5Cright%29%7D%20%5Cimplies%20f%27%3D%5Cfrac%7B1%7D%7Bx%7D%20" alt="LaTeX:  \displaystyle f=\log{\left(x \right)} \implies f'=\frac{1}{x} " data-equation-content=" \displaystyle f=\log{\left(x \right)} \implies f'=\frac{1}{x} " />  and  <img class="equation_image" title=" \displaystyle g=3 - 4 x \implies g'=-4 " src="/equation_images/%20%5Cdisplaystyle%20g%3D3%20-%204%20x%20%5Cimplies%20g%27%3D-4%20" alt="LaTeX:  \displaystyle g=3 - 4 x \implies g'=-4 " data-equation-content=" \displaystyle g=3 - 4 x \implies g'=-4 " /> . Popping up a level gives  <img class="equation_image" title=" \displaystyle g'=(3 - 4 x)(\frac{1}{x})+(\log{\left(x \right)})(-4) " src="/equation_images/%20%5Cdisplaystyle%20g%27%3D%283%20-%204%20x%29%28%5Cfrac%7B1%7D%7Bx%7D%29%2B%28%5Clog%7B%5Cleft%28x%20%5Cright%29%7D%29%28-4%29%20" alt="LaTeX:  \displaystyle g'=(3 - 4 x)(\frac{1}{x})+(\log{\left(x \right)})(-4) " data-equation-content=" \displaystyle g'=(3 - 4 x)(\frac{1}{x})+(\log{\left(x \right)})(-4) " /> Popping up again (Back to the original problem) gives  <img class="equation_image" title=" \displaystyle f'=(- 7 x - 9)(- 4 \log{\left(x \right)} + \frac{3 - 4 x}{x})+(\left(3 - 4 x\right) \log{\left(x \right)})(-7)=\left(28 x - 21\right) \log{\left(x \right)} + \left(28 x + 36\right) \log{\left(x \right)} + \frac{\left(3 - 4 x\right) \left(- 7 x - 9\right)}{x} " src="/equation_images/%20%5Cdisplaystyle%20f%27%3D%28-%207%20x%20-%209%29%28-%204%20%5Clog%7B%5Cleft%28x%20%5Cright%29%7D%20%2B%20%5Cfrac%7B3%20-%204%20x%7D%7Bx%7D%29%2B%28%5Cleft%283%20-%204%20x%5Cright%29%20%5Clog%7B%5Cleft%28x%20%5Cright%29%7D%29%28-7%29%3D%5Cleft%2828%20x%20-%2021%5Cright%29%20%5Clog%7B%5Cleft%28x%20%5Cright%29%7D%20%2B%20%5Cleft%2828%20x%20%2B%2036%5Cright%29%20%5Clog%7B%5Cleft%28x%20%5Cright%29%7D%20%2B%20%5Cfrac%7B%5Cleft%283%20-%204%20x%5Cright%29%20%5Cleft%28-%207%20x%20-%209%5Cright%29%7D%7Bx%7D%20" alt="LaTeX:  \displaystyle f'=(- 7 x - 9)(- 4 \log{\left(x \right)} + \frac{3 - 4 x}{x})+(\left(3 - 4 x\right) \log{\left(x \right)})(-7)=\left(28 x - 21\right) \log{\left(x \right)} + \left(28 x + 36\right) \log{\left(x \right)} + \frac{\left(3 - 4 x\right) \left(- 7 x - 9\right)}{x} " data-equation-content=" \displaystyle f'=(- 7 x - 9)(- 4 \log{\left(x \right)} + \frac{3 - 4 x}{x})+(\left(3 - 4 x\right) \log{\left(x \right)})(-7)=\left(28 x - 21\right) \log{\left(x \right)} + \left(28 x + 36\right) \log{\left(x \right)} + \frac{\left(3 - 4 x\right) \left(- 7 x - 9\right)}{x} " /> </p> </p>