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Identifying \(\displaystyle f=4 - 2 x\) and \(\displaystyle g=\left(5 x - 4\right) \log{\left(x \right)}\) and using the product rule with \(\displaystyle f=4 - 2 x \implies f'=-2\). This leaves g as \(\displaystyle g = \left(5 x - 4\right) \log{\left(x \right)}\) 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 \(\displaystyle g=5 x - 4 \implies g'=5\). Popping up a level gives \(\displaystyle g'=(5 x - 4)(\frac{1}{x})+(\log{\left(x \right)})(5)\)Popping up again (Back to the original problem) gives \(\displaystyle f'=(4 - 2 x)(5 \log{\left(x \right)} + \frac{5 x - 4}{x})+(\left(5 x - 4\right) \log{\left(x \right)})(-2)=\left(8 - 10 x\right) \log{\left(x \right)} + \left(20 - 10 x\right) \log{\left(x \right)} + \frac{\left(4 - 2 x\right) \left(5 x - 4\right)}{x}\)

Download \(\LaTeX\)

\begin{question}Find the derivative of $y = (4 - 2 x)(\log{\left(x \right)})(5 x - 4)$.
    \soln{9cm}{Identifying $f=4 - 2 x$ and $g=\left(5 x - 4\right) \log{\left(x \right)}$ and using the product rule with $f=4 - 2 x \implies f'=-2$. This leaves g as $g = \left(5 x - 4\right) \log{\left(x \right)}$ which also requires the product rule. Pushing down in the new product rule $f=\log{\left(x \right)} \implies f'=\frac{1}{x}$ and $g=5 x - 4 \implies g'=5$. Popping up a level gives $g'=(5 x - 4)(\frac{1}{x})+(\log{\left(x \right)})(5)$Popping up again (Back to the original problem) gives $f'=(4 - 2 x)(5 \log{\left(x \right)} + \frac{5 x - 4}{x})+(\left(5 x - 4\right) \log{\left(x \right)})(-2)=\left(8 - 10 x\right) \log{\left(x \right)} + \left(20 - 10 x\right) \log{\left(x \right)} + \frac{\left(4 - 2 x\right) \left(5 x - 4\right)}{x}$}

\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 = (4 - 2 x)(\log{\left(x \right)})(5 x - 4) " src="/equation_images/%20%5Cdisplaystyle%20y%20%3D%20%284%20-%202%20x%29%28%5Clog%7B%5Cleft%28x%20%5Cright%29%7D%29%285%20x%20-%204%29%20" alt="LaTeX:  \displaystyle y = (4 - 2 x)(\log{\left(x \right)})(5 x - 4) " data-equation-content=" \displaystyle y = (4 - 2 x)(\log{\left(x \right)})(5 x - 4) " /> .</p> </p>

HTML for Canvas

<p> <p>Identifying  <img class="equation_image" title=" \displaystyle f=4 - 2 x " src="/equation_images/%20%5Cdisplaystyle%20f%3D4%20-%202%20x%20" alt="LaTeX:  \displaystyle f=4 - 2 x " data-equation-content=" \displaystyle f=4 - 2 x " />  and  <img class="equation_image" title=" \displaystyle g=\left(5 x - 4\right) \log{\left(x \right)} " src="/equation_images/%20%5Cdisplaystyle%20g%3D%5Cleft%285%20x%20-%204%5Cright%29%20%5Clog%7B%5Cleft%28x%20%5Cright%29%7D%20" alt="LaTeX:  \displaystyle g=\left(5 x - 4\right) \log{\left(x \right)} " data-equation-content=" \displaystyle g=\left(5 x - 4\right) \log{\left(x \right)} " />  and using the product rule with  <img class="equation_image" title=" \displaystyle f=4 - 2 x \implies f'=-2 " src="/equation_images/%20%5Cdisplaystyle%20f%3D4%20-%202%20x%20%5Cimplies%20f%27%3D-2%20" alt="LaTeX:  \displaystyle f=4 - 2 x \implies f'=-2 " data-equation-content=" \displaystyle f=4 - 2 x \implies f'=-2 " /> . This leaves g as  <img class="equation_image" title=" \displaystyle g = \left(5 x - 4\right) \log{\left(x \right)} " src="/equation_images/%20%5Cdisplaystyle%20g%20%3D%20%5Cleft%285%20x%20-%204%5Cright%29%20%5Clog%7B%5Cleft%28x%20%5Cright%29%7D%20" alt="LaTeX:  \displaystyle g = \left(5 x - 4\right) \log{\left(x \right)} " data-equation-content=" \displaystyle g = \left(5 x - 4\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=5 x - 4 \implies g'=5 " src="/equation_images/%20%5Cdisplaystyle%20g%3D5%20x%20-%204%20%5Cimplies%20g%27%3D5%20" alt="LaTeX:  \displaystyle g=5 x - 4 \implies g'=5 " data-equation-content=" \displaystyle g=5 x - 4 \implies g'=5 " /> . Popping up a level gives  <img class="equation_image" title=" \displaystyle g'=(5 x - 4)(\frac{1}{x})+(\log{\left(x \right)})(5) " src="/equation_images/%20%5Cdisplaystyle%20g%27%3D%285%20x%20-%204%29%28%5Cfrac%7B1%7D%7Bx%7D%29%2B%28%5Clog%7B%5Cleft%28x%20%5Cright%29%7D%29%285%29%20" alt="LaTeX:  \displaystyle g'=(5 x - 4)(\frac{1}{x})+(\log{\left(x \right)})(5) " data-equation-content=" \displaystyle g'=(5 x - 4)(\frac{1}{x})+(\log{\left(x \right)})(5) " /> Popping up again (Back to the original problem) gives  <img class="equation_image" title=" \displaystyle f'=(4 - 2 x)(5 \log{\left(x \right)} + \frac{5 x - 4}{x})+(\left(5 x - 4\right) \log{\left(x \right)})(-2)=\left(8 - 10 x\right) \log{\left(x \right)} + \left(20 - 10 x\right) \log{\left(x \right)} + \frac{\left(4 - 2 x\right) \left(5 x - 4\right)}{x} " src="/equation_images/%20%5Cdisplaystyle%20f%27%3D%284%20-%202%20x%29%285%20%5Clog%7B%5Cleft%28x%20%5Cright%29%7D%20%2B%20%5Cfrac%7B5%20x%20-%204%7D%7Bx%7D%29%2B%28%5Cleft%285%20x%20-%204%5Cright%29%20%5Clog%7B%5Cleft%28x%20%5Cright%29%7D%29%28-2%29%3D%5Cleft%288%20-%2010%20x%5Cright%29%20%5Clog%7B%5Cleft%28x%20%5Cright%29%7D%20%2B%20%5Cleft%2820%20-%2010%20x%5Cright%29%20%5Clog%7B%5Cleft%28x%20%5Cright%29%7D%20%2B%20%5Cfrac%7B%5Cleft%284%20-%202%20x%5Cright%29%20%5Cleft%285%20x%20-%204%5Cright%29%7D%7Bx%7D%20" alt="LaTeX:  \displaystyle f'=(4 - 2 x)(5 \log{\left(x \right)} + \frac{5 x - 4}{x})+(\left(5 x - 4\right) \log{\left(x \right)})(-2)=\left(8 - 10 x\right) \log{\left(x \right)} + \left(20 - 10 x\right) \log{\left(x \right)} + \frac{\left(4 - 2 x\right) \left(5 x - 4\right)}{x} " data-equation-content=" \displaystyle f'=(4 - 2 x)(5 \log{\left(x \right)} + \frac{5 x - 4}{x})+(\left(5 x - 4\right) \log{\left(x \right)})(-2)=\left(8 - 10 x\right) \log{\left(x \right)} + \left(20 - 10 x\right) \log{\left(x \right)} + \frac{\left(4 - 2 x\right) \left(5 x - 4\right)}{x} " /> </p> </p>