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Find the derivative of \(\displaystyle y = (8 x^{3} - 5 x^{2} + 3 x + 6)(4 x^{3} - 4 x^{2} + 9 x - 2)(4 x^{3} + 3 x^{2} + 8 x - 7)\).
Identifying \(\displaystyle f=8 x^{3} - 5 x^{2} + 3 x + 6\) and \(\displaystyle g=\left(4 x^{3} - 4 x^{2} + 9 x - 2\right) \left(4 x^{3} + 3 x^{2} + 8 x - 7\right)\) and using the product rule with \(\displaystyle f=8 x^{3} - 5 x^{2} + 3 x + 6 \implies f'=24 x^{2} - 10 x + 3\). This leaves g as \(\displaystyle g = \left(4 x^{3} - 4 x^{2} + 9 x - 2\right) \left(4 x^{3} + 3 x^{2} + 8 x - 7\right)\) which also requires the product rule. Pushing down in the new product rule \(\displaystyle f=4 x^{3} - 4 x^{2} + 9 x - 2 \implies f'=12 x^{2} - 8 x + 9\) and \(\displaystyle g=4 x^{3} + 3 x^{2} + 8 x - 7 \implies g'=12 x^{2} + 6 x + 8\). Popping up a level gives \(\displaystyle g'=(4 x^{3} + 3 x^{2} + 8 x - 7)(12 x^{2} - 8 x + 9)+(4 x^{3} - 4 x^{2} + 9 x - 2)(12 x^{2} + 6 x + 8)\)Popping up again (Back to the original problem) gives \(\displaystyle f'=(8 x^{3} - 5 x^{2} + 3 x + 6)(\left(12 x^{2} - 8 x + 9\right) \left(4 x^{3} + 3 x^{2} + 8 x - 7\right) + \left(12 x^{2} + 6 x + 8\right) \left(4 x^{3} - 4 x^{2} + 9 x - 2\right))+(\left(4 x^{3} - 4 x^{2} + 9 x - 2\right) \left(4 x^{3} + 3 x^{2} + 8 x - 7\right))(24 x^{2} - 10 x + 3)=\left(12 x^{2} - 8 x + 9\right) \left(4 x^{3} + 3 x^{2} + 8 x - 7\right) \left(8 x^{3} - 5 x^{2} + 3 x + 6\right) + \left(12 x^{2} + 6 x + 8\right) \left(4 x^{3} - 4 x^{2} + 9 x - 2\right) \left(8 x^{3} - 5 x^{2} + 3 x + 6\right) + \left(24 x^{2} - 10 x + 3\right) \left(4 x^{3} - 4 x^{2} + 9 x - 2\right) \left(4 x^{3} + 3 x^{2} + 8 x - 7\right)\)
\begin{question}Find the derivative of $y = (8 x^{3} - 5 x^{2} + 3 x + 6)(4 x^{3} - 4 x^{2} + 9 x - 2)(4 x^{3} + 3 x^{2} + 8 x - 7)$.
\soln{9cm}{Identifying $f=8 x^{3} - 5 x^{2} + 3 x + 6$ and $g=\left(4 x^{3} - 4 x^{2} + 9 x - 2\right) \left(4 x^{3} + 3 x^{2} + 8 x - 7\right)$ and using the product rule with $f=8 x^{3} - 5 x^{2} + 3 x + 6 \implies f'=24 x^{2} - 10 x + 3$. This leaves g as $g = \left(4 x^{3} - 4 x^{2} + 9 x - 2\right) \left(4 x^{3} + 3 x^{2} + 8 x - 7\right)$ which also requires the product rule. Pushing down in the new product rule $f=4 x^{3} - 4 x^{2} + 9 x - 2 \implies f'=12 x^{2} - 8 x + 9$ and $g=4 x^{3} + 3 x^{2} + 8 x - 7 \implies g'=12 x^{2} + 6 x + 8$. Popping up a level gives $g'=(4 x^{3} + 3 x^{2} + 8 x - 7)(12 x^{2} - 8 x + 9)+(4 x^{3} - 4 x^{2} + 9 x - 2)(12 x^{2} + 6 x + 8)$Popping up again (Back to the original problem) gives $f'=(8 x^{3} - 5 x^{2} + 3 x + 6)(\left(12 x^{2} - 8 x + 9\right) \left(4 x^{3} + 3 x^{2} + 8 x - 7\right) + \left(12 x^{2} + 6 x + 8\right) \left(4 x^{3} - 4 x^{2} + 9 x - 2\right))+(\left(4 x^{3} - 4 x^{2} + 9 x - 2\right) \left(4 x^{3} + 3 x^{2} + 8 x - 7\right))(24 x^{2} - 10 x + 3)=\left(12 x^{2} - 8 x + 9\right) \left(4 x^{3} + 3 x^{2} + 8 x - 7\right) \left(8 x^{3} - 5 x^{2} + 3 x + 6\right) + \left(12 x^{2} + 6 x + 8\right) \left(4 x^{3} - 4 x^{2} + 9 x - 2\right) \left(8 x^{3} - 5 x^{2} + 3 x + 6\right) + \left(24 x^{2} - 10 x + 3\right) \left(4 x^{3} - 4 x^{2} + 9 x - 2\right) \left(4 x^{3} + 3 x^{2} + 8 x - 7\right)$}
\end{question}
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\begin{document}\begin{question}(10pts) The question goes here!
\soln{9cm}{The solution goes here.}
\end{question}\end{document}<p> <p>Find the derivative of <img class="equation_image" title=" \displaystyle y = (8 x^{3} - 5 x^{2} + 3 x + 6)(4 x^{3} - 4 x^{2} + 9 x - 2)(4 x^{3} + 3 x^{2} + 8 x - 7) " src="/equation_images/%20%5Cdisplaystyle%20y%20%3D%20%288%20x%5E%7B3%7D%20-%205%20x%5E%7B2%7D%20%2B%203%20x%20%2B%206%29%284%20x%5E%7B3%7D%20-%204%20x%5E%7B2%7D%20%2B%209%20x%20-%202%29%284%20x%5E%7B3%7D%20%2B%203%20x%5E%7B2%7D%20%2B%208%20x%20-%207%29%20" alt="LaTeX: \displaystyle y = (8 x^{3} - 5 x^{2} + 3 x + 6)(4 x^{3} - 4 x^{2} + 9 x - 2)(4 x^{3} + 3 x^{2} + 8 x - 7) " data-equation-content=" \displaystyle y = (8 x^{3} - 5 x^{2} + 3 x + 6)(4 x^{3} - 4 x^{2} + 9 x - 2)(4 x^{3} + 3 x^{2} + 8 x - 7) " /> .</p> </p><p> <p>Identifying <img class="equation_image" title=" \displaystyle f=8 x^{3} - 5 x^{2} + 3 x + 6 " src="/equation_images/%20%5Cdisplaystyle%20f%3D8%20x%5E%7B3%7D%20-%205%20x%5E%7B2%7D%20%2B%203%20x%20%2B%206%20" alt="LaTeX: \displaystyle f=8 x^{3} - 5 x^{2} + 3 x + 6 " data-equation-content=" \displaystyle f=8 x^{3} - 5 x^{2} + 3 x + 6 " /> and <img class="equation_image" title=" \displaystyle g=\left(4 x^{3} - 4 x^{2} + 9 x - 2\right) \left(4 x^{3} + 3 x^{2} + 8 x - 7\right) " src="/equation_images/%20%5Cdisplaystyle%20g%3D%5Cleft%284%20x%5E%7B3%7D%20-%204%20x%5E%7B2%7D%20%2B%209%20x%20-%202%5Cright%29%20%5Cleft%284%20x%5E%7B3%7D%20%2B%203%20x%5E%7B2%7D%20%2B%208%20x%20-%207%5Cright%29%20" alt="LaTeX: \displaystyle g=\left(4 x^{3} - 4 x^{2} + 9 x - 2\right) \left(4 x^{3} + 3 x^{2} + 8 x - 7\right) " data-equation-content=" \displaystyle g=\left(4 x^{3} - 4 x^{2} + 9 x - 2\right) \left(4 x^{3} + 3 x^{2} + 8 x - 7\right) " /> and using the product rule with <img class="equation_image" title=" \displaystyle f=8 x^{3} - 5 x^{2} + 3 x + 6 \implies f'=24 x^{2} - 10 x + 3 " src="/equation_images/%20%5Cdisplaystyle%20f%3D8%20x%5E%7B3%7D%20-%205%20x%5E%7B2%7D%20%2B%203%20x%20%2B%206%20%5Cimplies%20f%27%3D24%20x%5E%7B2%7D%20-%2010%20x%20%2B%203%20" alt="LaTeX: \displaystyle f=8 x^{3} - 5 x^{2} + 3 x + 6 \implies f'=24 x^{2} - 10 x + 3 " data-equation-content=" \displaystyle f=8 x^{3} - 5 x^{2} + 3 x + 6 \implies f'=24 x^{2} - 10 x + 3 " /> . This leaves g as <img class="equation_image" title=" \displaystyle g = \left(4 x^{3} - 4 x^{2} + 9 x - 2\right) \left(4 x^{3} + 3 x^{2} + 8 x - 7\right) " src="/equation_images/%20%5Cdisplaystyle%20g%20%3D%20%5Cleft%284%20x%5E%7B3%7D%20-%204%20x%5E%7B2%7D%20%2B%209%20x%20-%202%5Cright%29%20%5Cleft%284%20x%5E%7B3%7D%20%2B%203%20x%5E%7B2%7D%20%2B%208%20x%20-%207%5Cright%29%20" alt="LaTeX: \displaystyle g = \left(4 x^{3} - 4 x^{2} + 9 x - 2\right) \left(4 x^{3} + 3 x^{2} + 8 x - 7\right) " data-equation-content=" \displaystyle g = \left(4 x^{3} - 4 x^{2} + 9 x - 2\right) \left(4 x^{3} + 3 x^{2} + 8 x - 7\right) " /> which also requires the product rule. Pushing down in the new product rule <img class="equation_image" title=" \displaystyle f=4 x^{3} - 4 x^{2} + 9 x - 2 \implies f'=12 x^{2} - 8 x + 9 " src="/equation_images/%20%5Cdisplaystyle%20f%3D4%20x%5E%7B3%7D%20-%204%20x%5E%7B2%7D%20%2B%209%20x%20-%202%20%5Cimplies%20f%27%3D12%20x%5E%7B2%7D%20-%208%20x%20%2B%209%20" alt="LaTeX: \displaystyle f=4 x^{3} - 4 x^{2} + 9 x - 2 \implies f'=12 x^{2} - 8 x + 9 " data-equation-content=" \displaystyle f=4 x^{3} - 4 x^{2} + 9 x - 2 \implies f'=12 x^{2} - 8 x + 9 " /> and <img class="equation_image" title=" \displaystyle g=4 x^{3} + 3 x^{2} + 8 x - 7 \implies g'=12 x^{2} + 6 x + 8 " src="/equation_images/%20%5Cdisplaystyle%20g%3D4%20x%5E%7B3%7D%20%2B%203%20x%5E%7B2%7D%20%2B%208%20x%20-%207%20%5Cimplies%20g%27%3D12%20x%5E%7B2%7D%20%2B%206%20x%20%2B%208%20" alt="LaTeX: \displaystyle g=4 x^{3} + 3 x^{2} + 8 x - 7 \implies g'=12 x^{2} + 6 x + 8 " data-equation-content=" \displaystyle g=4 x^{3} + 3 x^{2} + 8 x - 7 \implies g'=12 x^{2} + 6 x + 8 " /> . Popping up a level gives <img class="equation_image" title=" \displaystyle g'=(4 x^{3} + 3 x^{2} + 8 x - 7)(12 x^{2} - 8 x + 9)+(4 x^{3} - 4 x^{2} + 9 x - 2)(12 x^{2} + 6 x + 8) " src="/equation_images/%20%5Cdisplaystyle%20g%27%3D%284%20x%5E%7B3%7D%20%2B%203%20x%5E%7B2%7D%20%2B%208%20x%20-%207%29%2812%20x%5E%7B2%7D%20-%208%20x%20%2B%209%29%2B%284%20x%5E%7B3%7D%20-%204%20x%5E%7B2%7D%20%2B%209%20x%20-%202%29%2812%20x%5E%7B2%7D%20%2B%206%20x%20%2B%208%29%20" alt="LaTeX: \displaystyle g'=(4 x^{3} + 3 x^{2} + 8 x - 7)(12 x^{2} - 8 x + 9)+(4 x^{3} - 4 x^{2} + 9 x - 2)(12 x^{2} + 6 x + 8) " data-equation-content=" \displaystyle g'=(4 x^{3} + 3 x^{2} + 8 x - 7)(12 x^{2} - 8 x + 9)+(4 x^{3} - 4 x^{2} + 9 x - 2)(12 x^{2} + 6 x + 8) " /> Popping up again (Back to the original problem) gives <img class="equation_image" title=" \displaystyle f'=(8 x^{3} - 5 x^{2} + 3 x + 6)(\left(12 x^{2} - 8 x + 9\right) \left(4 x^{3} + 3 x^{2} + 8 x - 7\right) + \left(12 x^{2} + 6 x + 8\right) \left(4 x^{3} - 4 x^{2} + 9 x - 2\right))+(\left(4 x^{3} - 4 x^{2} + 9 x - 2\right) \left(4 x^{3} + 3 x^{2} + 8 x - 7\right))(24 x^{2} - 10 x + 3)=\left(12 x^{2} - 8 x + 9\right) \left(4 x^{3} + 3 x^{2} + 8 x - 7\right) \left(8 x^{3} - 5 x^{2} + 3 x + 6\right) + \left(12 x^{2} + 6 x + 8\right) \left(4 x^{3} - 4 x^{2} + 9 x - 2\right) \left(8 x^{3} - 5 x^{2} + 3 x + 6\right) + \left(24 x^{2} - 10 x + 3\right) \left(4 x^{3} - 4 x^{2} + 9 x - 2\right) \left(4 x^{3} + 3 x^{2} + 8 x - 7\right) " src="/equation_images/%20%5Cdisplaystyle%20f%27%3D%288%20x%5E%7B3%7D%20-%205%20x%5E%7B2%7D%20%2B%203%20x%20%2B%206%29%28%5Cleft%2812%20x%5E%7B2%7D%20-%208%20x%20%2B%209%5Cright%29%20%5Cleft%284%20x%5E%7B3%7D%20%2B%203%20x%5E%7B2%7D%20%2B%208%20x%20-%207%5Cright%29%20%2B%20%5Cleft%2812%20x%5E%7B2%7D%20%2B%206%20x%20%2B%208%5Cright%29%20%5Cleft%284%20x%5E%7B3%7D%20-%204%20x%5E%7B2%7D%20%2B%209%20x%20-%202%5Cright%29%29%2B%28%5Cleft%284%20x%5E%7B3%7D%20-%204%20x%5E%7B2%7D%20%2B%209%20x%20-%202%5Cright%29%20%5Cleft%284%20x%5E%7B3%7D%20%2B%203%20x%5E%7B2%7D%20%2B%208%20x%20-%207%5Cright%29%29%2824%20x%5E%7B2%7D%20-%2010%20x%20%2B%203%29%3D%5Cleft%2812%20x%5E%7B2%7D%20-%208%20x%20%2B%209%5Cright%29%20%5Cleft%284%20x%5E%7B3%7D%20%2B%203%20x%5E%7B2%7D%20%2B%208%20x%20-%207%5Cright%29%20%5Cleft%288%20x%5E%7B3%7D%20-%205%20x%5E%7B2%7D%20%2B%203%20x%20%2B%206%5Cright%29%20%2B%20%5Cleft%2812%20x%5E%7B2%7D%20%2B%206%20x%20%2B%208%5Cright%29%20%5Cleft%284%20x%5E%7B3%7D%20-%204%20x%5E%7B2%7D%20%2B%209%20x%20-%202%5Cright%29%20%5Cleft%288%20x%5E%7B3%7D%20-%205%20x%5E%7B2%7D%20%2B%203%20x%20%2B%206%5Cright%29%20%2B%20%5Cleft%2824%20x%5E%7B2%7D%20-%2010%20x%20%2B%203%5Cright%29%20%5Cleft%284%20x%5E%7B3%7D%20-%204%20x%5E%7B2%7D%20%2B%209%20x%20-%202%5Cright%29%20%5Cleft%284%20x%5E%7B3%7D%20%2B%203%20x%5E%7B2%7D%20%2B%208%20x%20-%207%5Cright%29%20" alt="LaTeX: \displaystyle f'=(8 x^{3} - 5 x^{2} + 3 x + 6)(\left(12 x^{2} - 8 x + 9\right) \left(4 x^{3} + 3 x^{2} + 8 x - 7\right) + \left(12 x^{2} + 6 x + 8\right) \left(4 x^{3} - 4 x^{2} + 9 x - 2\right))+(\left(4 x^{3} - 4 x^{2} + 9 x - 2\right) \left(4 x^{3} + 3 x^{2} + 8 x - 7\right))(24 x^{2} - 10 x + 3)=\left(12 x^{2} - 8 x + 9\right) \left(4 x^{3} + 3 x^{2} + 8 x - 7\right) \left(8 x^{3} - 5 x^{2} + 3 x + 6\right) + \left(12 x^{2} + 6 x + 8\right) \left(4 x^{3} - 4 x^{2} + 9 x - 2\right) \left(8 x^{3} - 5 x^{2} + 3 x + 6\right) + \left(24 x^{2} - 10 x + 3\right) \left(4 x^{3} - 4 x^{2} + 9 x - 2\right) \left(4 x^{3} + 3 x^{2} + 8 x - 7\right) " data-equation-content=" \displaystyle f'=(8 x^{3} - 5 x^{2} + 3 x + 6)(\left(12 x^{2} - 8 x + 9\right) \left(4 x^{3} + 3 x^{2} + 8 x - 7\right) + \left(12 x^{2} + 6 x + 8\right) \left(4 x^{3} - 4 x^{2} + 9 x - 2\right))+(\left(4 x^{3} - 4 x^{2} + 9 x - 2\right) \left(4 x^{3} + 3 x^{2} + 8 x - 7\right))(24 x^{2} - 10 x + 3)=\left(12 x^{2} - 8 x + 9\right) \left(4 x^{3} + 3 x^{2} + 8 x - 7\right) \left(8 x^{3} - 5 x^{2} + 3 x + 6\right) + \left(12 x^{2} + 6 x + 8\right) \left(4 x^{3} - 4 x^{2} + 9 x - 2\right) \left(8 x^{3} - 5 x^{2} + 3 x + 6\right) + \left(24 x^{2} - 10 x + 3\right) \left(4 x^{3} - 4 x^{2} + 9 x - 2\right) \left(4 x^{3} + 3 x^{2} + 8 x - 7\right) " /> </p> </p>