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Identifying \(\displaystyle f=- 6 x - 1\) and \(\displaystyle g=\left(- 4 x - 1\right) \left(7 x + 5\right)\) and using the product rule with \(\displaystyle f=- 6 x - 1 \implies f'=-6\). This leaves g as \(\displaystyle g = \left(- 4 x - 1\right) \left(7 x + 5\right)\) which also requires the product rule. Pushing down in the new product rule \(\displaystyle f=7 x + 5 \implies f'=7\) and \(\displaystyle g=- 4 x - 1 \implies g'=-4\). Popping up a level gives \(\displaystyle g'=(- 4 x - 1)(7)+(7 x + 5)(-4)\)Popping up again (Back to the original problem) gives \(\displaystyle f'=(- 6 x - 1)(- 56 x - 27)+(\left(- 4 x - 1\right) \left(7 x + 5\right))(-6)=7 \left(- 6 x - 1\right) \left(- 4 x - 1\right) + \left(7 x + 5\right) \left(24 x + 4\right) + \left(7 x + 5\right) \left(24 x + 6\right)\)

Download \(\LaTeX\)

\begin{question}Find the derivative of $y = (- 6 x - 1)(7 x + 5)(- 4 x - 1)$.
    \soln{9cm}{Identifying $f=- 6 x - 1$ and $g=\left(- 4 x - 1\right) \left(7 x + 5\right)$ and using the product rule with $f=- 6 x - 1 \implies f'=-6$. This leaves g as $g = \left(- 4 x - 1\right) \left(7 x + 5\right)$ which also requires the product rule. Pushing down in the new product rule $f=7 x + 5 \implies f'=7$ and $g=- 4 x - 1 \implies g'=-4$. Popping up a level gives $g'=(- 4 x - 1)(7)+(7 x + 5)(-4)$Popping up again (Back to the original problem) gives $f'=(- 6 x - 1)(- 56 x - 27)+(\left(- 4 x - 1\right) \left(7 x + 5\right))(-6)=7 \left(- 6 x - 1\right) \left(- 4 x - 1\right) + \left(7 x + 5\right) \left(24 x + 4\right) + \left(7 x + 5\right) \left(24 x + 6\right)$}

\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>Find the derivative of  <img class="equation_image" title=" \displaystyle y = (- 6 x - 1)(7 x + 5)(- 4 x - 1) " src="/equation_images/%20%5Cdisplaystyle%20y%20%3D%20%28-%206%20x%20-%201%29%287%20x%20%2B%205%29%28-%204%20x%20-%201%29%20" alt="LaTeX:  \displaystyle y = (- 6 x - 1)(7 x + 5)(- 4 x - 1) " data-equation-content=" \displaystyle y = (- 6 x - 1)(7 x + 5)(- 4 x - 1) " /> .</p> </p>

HTML for Canvas

<p> <p>Identifying  <img class="equation_image" title=" \displaystyle f=- 6 x - 1 " src="/equation_images/%20%5Cdisplaystyle%20f%3D-%206%20x%20-%201%20" alt="LaTeX:  \displaystyle f=- 6 x - 1 " data-equation-content=" \displaystyle f=- 6 x - 1 " />  and  <img class="equation_image" title=" \displaystyle g=\left(- 4 x - 1\right) \left(7 x + 5\right) " src="/equation_images/%20%5Cdisplaystyle%20g%3D%5Cleft%28-%204%20x%20-%201%5Cright%29%20%5Cleft%287%20x%20%2B%205%5Cright%29%20" alt="LaTeX:  \displaystyle g=\left(- 4 x - 1\right) \left(7 x + 5\right) " data-equation-content=" \displaystyle g=\left(- 4 x - 1\right) \left(7 x + 5\right) " />  and using the product rule with  <img class="equation_image" title=" \displaystyle f=- 6 x - 1 \implies f'=-6 " src="/equation_images/%20%5Cdisplaystyle%20f%3D-%206%20x%20-%201%20%5Cimplies%20f%27%3D-6%20" alt="LaTeX:  \displaystyle f=- 6 x - 1 \implies f'=-6 " data-equation-content=" \displaystyle f=- 6 x - 1 \implies f'=-6 " /> . This leaves g as  <img class="equation_image" title=" \displaystyle g = \left(- 4 x - 1\right) \left(7 x + 5\right) " src="/equation_images/%20%5Cdisplaystyle%20g%20%3D%20%5Cleft%28-%204%20x%20-%201%5Cright%29%20%5Cleft%287%20x%20%2B%205%5Cright%29%20" alt="LaTeX:  \displaystyle g = \left(- 4 x - 1\right) \left(7 x + 5\right) " data-equation-content=" \displaystyle g = \left(- 4 x - 1\right) \left(7 x + 5\right) " />  which also requires the product rule. Pushing down in the new product rule  <img class="equation_image" title=" \displaystyle f=7 x + 5 \implies f'=7 " src="/equation_images/%20%5Cdisplaystyle%20f%3D7%20x%20%2B%205%20%5Cimplies%20f%27%3D7%20" alt="LaTeX:  \displaystyle f=7 x + 5 \implies f'=7 " data-equation-content=" \displaystyle f=7 x + 5 \implies f'=7 " />  and  <img class="equation_image" title=" \displaystyle g=- 4 x - 1 \implies g'=-4 " src="/equation_images/%20%5Cdisplaystyle%20g%3D-%204%20x%20-%201%20%5Cimplies%20g%27%3D-4%20" alt="LaTeX:  \displaystyle g=- 4 x - 1 \implies g'=-4 " data-equation-content=" \displaystyle g=- 4 x - 1 \implies g'=-4 " /> . Popping up a level gives  <img class="equation_image" title=" \displaystyle g'=(- 4 x - 1)(7)+(7 x + 5)(-4) " src="/equation_images/%20%5Cdisplaystyle%20g%27%3D%28-%204%20x%20-%201%29%287%29%2B%287%20x%20%2B%205%29%28-4%29%20" alt="LaTeX:  \displaystyle g'=(- 4 x - 1)(7)+(7 x + 5)(-4) " data-equation-content=" \displaystyle g'=(- 4 x - 1)(7)+(7 x + 5)(-4) " /> Popping up again (Back to the original problem) gives  <img class="equation_image" title=" \displaystyle f'=(- 6 x - 1)(- 56 x - 27)+(\left(- 4 x - 1\right) \left(7 x + 5\right))(-6)=7 \left(- 6 x - 1\right) \left(- 4 x - 1\right) + \left(7 x + 5\right) \left(24 x + 4\right) + \left(7 x + 5\right) \left(24 x + 6\right) " src="/equation_images/%20%5Cdisplaystyle%20f%27%3D%28-%206%20x%20-%201%29%28-%2056%20x%20-%2027%29%2B%28%5Cleft%28-%204%20x%20-%201%5Cright%29%20%5Cleft%287%20x%20%2B%205%5Cright%29%29%28-6%29%3D7%20%5Cleft%28-%206%20x%20-%201%5Cright%29%20%5Cleft%28-%204%20x%20-%201%5Cright%29%20%2B%20%5Cleft%287%20x%20%2B%205%5Cright%29%20%5Cleft%2824%20x%20%2B%204%5Cright%29%20%2B%20%5Cleft%287%20x%20%2B%205%5Cright%29%20%5Cleft%2824%20x%20%2B%206%5Cright%29%20" alt="LaTeX:  \displaystyle f'=(- 6 x - 1)(- 56 x - 27)+(\left(- 4 x - 1\right) \left(7 x + 5\right))(-6)=7 \left(- 6 x - 1\right) \left(- 4 x - 1\right) + \left(7 x + 5\right) \left(24 x + 4\right) + \left(7 x + 5\right) \left(24 x + 6\right) " data-equation-content=" \displaystyle f'=(- 6 x - 1)(- 56 x - 27)+(\left(- 4 x - 1\right) \left(7 x + 5\right))(-6)=7 \left(- 6 x - 1\right) \left(- 4 x - 1\right) + \left(7 x + 5\right) \left(24 x + 4\right) + \left(7 x + 5\right) \left(24 x + 6\right) " /> </p> </p>