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Find the derivative of \(\displaystyle y = (8 x^{3} - 2 x^{2} + 4 x - 7)(9 x^{3} + 2 x^{2} - 5 x - 7)(- 4 x^{3} - x^{2} - 9 x + 7)\).
Identifying \(\displaystyle f=8 x^{3} - 2 x^{2} + 4 x - 7\) and \(\displaystyle g=\left(- 4 x^{3} - x^{2} - 9 x + 7\right) \left(9 x^{3} + 2 x^{2} - 5 x - 7\right)\) and using the product rule with \(\displaystyle f=8 x^{3} - 2 x^{2} + 4 x - 7 \implies f'=24 x^{2} - 4 x + 4\). This leaves g as \(\displaystyle g = \left(- 4 x^{3} - x^{2} - 9 x + 7\right) \left(9 x^{3} + 2 x^{2} - 5 x - 7\right)\) which also requires the product rule. Pushing down in the new product rule \(\displaystyle f=9 x^{3} + 2 x^{2} - 5 x - 7 \implies f'=27 x^{2} + 4 x - 5\) and \(\displaystyle g=- 4 x^{3} - x^{2} - 9 x + 7 \implies g'=- 12 x^{2} - 2 x - 9\). Popping up a level gives \(\displaystyle g'=(- 4 x^{3} - x^{2} - 9 x + 7)(27 x^{2} + 4 x - 5)+(9 x^{3} + 2 x^{2} - 5 x - 7)(- 12 x^{2} - 2 x - 9)\)Popping up again (Back to the original problem) gives \(\displaystyle f'=(8 x^{3} - 2 x^{2} + 4 x - 7)(\left(- 12 x^{2} - 2 x - 9\right) \left(9 x^{3} + 2 x^{2} - 5 x - 7\right) + \left(27 x^{2} + 4 x - 5\right) \left(- 4 x^{3} - x^{2} - 9 x + 7\right))+(\left(- 4 x^{3} - x^{2} - 9 x + 7\right) \left(9 x^{3} + 2 x^{2} - 5 x - 7\right))(24 x^{2} - 4 x + 4)=\left(- 12 x^{2} - 2 x - 9\right) \left(8 x^{3} - 2 x^{2} + 4 x - 7\right) \left(9 x^{3} + 2 x^{2} - 5 x - 7\right) + \left(24 x^{2} - 4 x + 4\right) \left(- 4 x^{3} - x^{2} - 9 x + 7\right) \left(9 x^{3} + 2 x^{2} - 5 x - 7\right) + \left(27 x^{2} + 4 x - 5\right) \left(- 4 x^{3} - x^{2} - 9 x + 7\right) \left(8 x^{3} - 2 x^{2} + 4 x - 7\right)\)
\begin{question}Find the derivative of $y = (8 x^{3} - 2 x^{2} + 4 x - 7)(9 x^{3} + 2 x^{2} - 5 x - 7)(- 4 x^{3} - x^{2} - 9 x + 7)$.
\soln{9cm}{Identifying $f=8 x^{3} - 2 x^{2} + 4 x - 7$ and $g=\left(- 4 x^{3} - x^{2} - 9 x + 7\right) \left(9 x^{3} + 2 x^{2} - 5 x - 7\right)$ and using the product rule with $f=8 x^{3} - 2 x^{2} + 4 x - 7 \implies f'=24 x^{2} - 4 x + 4$. This leaves g as $g = \left(- 4 x^{3} - x^{2} - 9 x + 7\right) \left(9 x^{3} + 2 x^{2} - 5 x - 7\right)$ which also requires the product rule. Pushing down in the new product rule $f=9 x^{3} + 2 x^{2} - 5 x - 7 \implies f'=27 x^{2} + 4 x - 5$ and $g=- 4 x^{3} - x^{2} - 9 x + 7 \implies g'=- 12 x^{2} - 2 x - 9$. Popping up a level gives $g'=(- 4 x^{3} - x^{2} - 9 x + 7)(27 x^{2} + 4 x - 5)+(9 x^{3} + 2 x^{2} - 5 x - 7)(- 12 x^{2} - 2 x - 9)$Popping up again (Back to the original problem) gives $f'=(8 x^{3} - 2 x^{2} + 4 x - 7)(\left(- 12 x^{2} - 2 x - 9\right) \left(9 x^{3} + 2 x^{2} - 5 x - 7\right) + \left(27 x^{2} + 4 x - 5\right) \left(- 4 x^{3} - x^{2} - 9 x + 7\right))+(\left(- 4 x^{3} - x^{2} - 9 x + 7\right) \left(9 x^{3} + 2 x^{2} - 5 x - 7\right))(24 x^{2} - 4 x + 4)=\left(- 12 x^{2} - 2 x - 9\right) \left(8 x^{3} - 2 x^{2} + 4 x - 7\right) \left(9 x^{3} + 2 x^{2} - 5 x - 7\right) + \left(24 x^{2} - 4 x + 4\right) \left(- 4 x^{3} - x^{2} - 9 x + 7\right) \left(9 x^{3} + 2 x^{2} - 5 x - 7\right) + \left(27 x^{2} + 4 x - 5\right) \left(- 4 x^{3} - x^{2} - 9 x + 7\right) \left(8 x^{3} - 2 x^{2} + 4 x - 7\right)$}
\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} - 2 x^{2} + 4 x - 7)(9 x^{3} + 2 x^{2} - 5 x - 7)(- 4 x^{3} - x^{2} - 9 x + 7) " src="/equation_images/%20%5Cdisplaystyle%20y%20%3D%20%288%20x%5E%7B3%7D%20-%202%20x%5E%7B2%7D%20%2B%204%20x%20-%207%29%289%20x%5E%7B3%7D%20%2B%202%20x%5E%7B2%7D%20-%205%20x%20-%207%29%28-%204%20x%5E%7B3%7D%20-%20x%5E%7B2%7D%20-%209%20x%20%2B%207%29%20" alt="LaTeX: \displaystyle y = (8 x^{3} - 2 x^{2} + 4 x - 7)(9 x^{3} + 2 x^{2} - 5 x - 7)(- 4 x^{3} - x^{2} - 9 x + 7) " data-equation-content=" \displaystyle y = (8 x^{3} - 2 x^{2} + 4 x - 7)(9 x^{3} + 2 x^{2} - 5 x - 7)(- 4 x^{3} - x^{2} - 9 x + 7) " /> .</p> </p><p> <p>Identifying <img class="equation_image" title=" \displaystyle f=8 x^{3} - 2 x^{2} + 4 x - 7 " src="/equation_images/%20%5Cdisplaystyle%20f%3D8%20x%5E%7B3%7D%20-%202%20x%5E%7B2%7D%20%2B%204%20x%20-%207%20" alt="LaTeX: \displaystyle f=8 x^{3} - 2 x^{2} + 4 x - 7 " data-equation-content=" \displaystyle f=8 x^{3} - 2 x^{2} + 4 x - 7 " /> and <img class="equation_image" title=" \displaystyle g=\left(- 4 x^{3} - x^{2} - 9 x + 7\right) \left(9 x^{3} + 2 x^{2} - 5 x - 7\right) " src="/equation_images/%20%5Cdisplaystyle%20g%3D%5Cleft%28-%204%20x%5E%7B3%7D%20-%20x%5E%7B2%7D%20-%209%20x%20%2B%207%5Cright%29%20%5Cleft%289%20x%5E%7B3%7D%20%2B%202%20x%5E%7B2%7D%20-%205%20x%20-%207%5Cright%29%20" alt="LaTeX: \displaystyle g=\left(- 4 x^{3} - x^{2} - 9 x + 7\right) \left(9 x^{3} + 2 x^{2} - 5 x - 7\right) " data-equation-content=" \displaystyle g=\left(- 4 x^{3} - x^{2} - 9 x + 7\right) \left(9 x^{3} + 2 x^{2} - 5 x - 7\right) " /> and using the product rule with <img class="equation_image" title=" \displaystyle f=8 x^{3} - 2 x^{2} + 4 x - 7 \implies f'=24 x^{2} - 4 x + 4 " src="/equation_images/%20%5Cdisplaystyle%20f%3D8%20x%5E%7B3%7D%20-%202%20x%5E%7B2%7D%20%2B%204%20x%20-%207%20%5Cimplies%20f%27%3D24%20x%5E%7B2%7D%20-%204%20x%20%2B%204%20" alt="LaTeX: \displaystyle f=8 x^{3} - 2 x^{2} + 4 x - 7 \implies f'=24 x^{2} - 4 x + 4 " data-equation-content=" \displaystyle f=8 x^{3} - 2 x^{2} + 4 x - 7 \implies f'=24 x^{2} - 4 x + 4 " /> . This leaves g as <img class="equation_image" title=" \displaystyle g = \left(- 4 x^{3} - x^{2} - 9 x + 7\right) \left(9 x^{3} + 2 x^{2} - 5 x - 7\right) " src="/equation_images/%20%5Cdisplaystyle%20g%20%3D%20%5Cleft%28-%204%20x%5E%7B3%7D%20-%20x%5E%7B2%7D%20-%209%20x%20%2B%207%5Cright%29%20%5Cleft%289%20x%5E%7B3%7D%20%2B%202%20x%5E%7B2%7D%20-%205%20x%20-%207%5Cright%29%20" alt="LaTeX: \displaystyle g = \left(- 4 x^{3} - x^{2} - 9 x + 7\right) \left(9 x^{3} + 2 x^{2} - 5 x - 7\right) " data-equation-content=" \displaystyle g = \left(- 4 x^{3} - x^{2} - 9 x + 7\right) \left(9 x^{3} + 2 x^{2} - 5 x - 7\right) " /> which also requires the product rule. Pushing down in the new product rule <img class="equation_image" title=" \displaystyle f=9 x^{3} + 2 x^{2} - 5 x - 7 \implies f'=27 x^{2} + 4 x - 5 " src="/equation_images/%20%5Cdisplaystyle%20f%3D9%20x%5E%7B3%7D%20%2B%202%20x%5E%7B2%7D%20-%205%20x%20-%207%20%5Cimplies%20f%27%3D27%20x%5E%7B2%7D%20%2B%204%20x%20-%205%20" alt="LaTeX: \displaystyle f=9 x^{3} + 2 x^{2} - 5 x - 7 \implies f'=27 x^{2} + 4 x - 5 " data-equation-content=" \displaystyle f=9 x^{3} + 2 x^{2} - 5 x - 7 \implies f'=27 x^{2} + 4 x - 5 " /> and <img class="equation_image" title=" \displaystyle g=- 4 x^{3} - x^{2} - 9 x + 7 \implies g'=- 12 x^{2} - 2 x - 9 " src="/equation_images/%20%5Cdisplaystyle%20g%3D-%204%20x%5E%7B3%7D%20-%20x%5E%7B2%7D%20-%209%20x%20%2B%207%20%5Cimplies%20g%27%3D-%2012%20x%5E%7B2%7D%20-%202%20x%20-%209%20" alt="LaTeX: \displaystyle g=- 4 x^{3} - x^{2} - 9 x + 7 \implies g'=- 12 x^{2} - 2 x - 9 " data-equation-content=" \displaystyle g=- 4 x^{3} - x^{2} - 9 x + 7 \implies g'=- 12 x^{2} - 2 x - 9 " /> . Popping up a level gives <img class="equation_image" title=" \displaystyle g'=(- 4 x^{3} - x^{2} - 9 x + 7)(27 x^{2} + 4 x - 5)+(9 x^{3} + 2 x^{2} - 5 x - 7)(- 12 x^{2} - 2 x - 9) " src="/equation_images/%20%5Cdisplaystyle%20g%27%3D%28-%204%20x%5E%7B3%7D%20-%20x%5E%7B2%7D%20-%209%20x%20%2B%207%29%2827%20x%5E%7B2%7D%20%2B%204%20x%20-%205%29%2B%289%20x%5E%7B3%7D%20%2B%202%20x%5E%7B2%7D%20-%205%20x%20-%207%29%28-%2012%20x%5E%7B2%7D%20-%202%20x%20-%209%29%20" alt="LaTeX: \displaystyle g'=(- 4 x^{3} - x^{2} - 9 x + 7)(27 x^{2} + 4 x - 5)+(9 x^{3} + 2 x^{2} - 5 x - 7)(- 12 x^{2} - 2 x - 9) " data-equation-content=" \displaystyle g'=(- 4 x^{3} - x^{2} - 9 x + 7)(27 x^{2} + 4 x - 5)+(9 x^{3} + 2 x^{2} - 5 x - 7)(- 12 x^{2} - 2 x - 9) " /> Popping up again (Back to the original problem) gives <img class="equation_image" title=" \displaystyle f'=(8 x^{3} - 2 x^{2} + 4 x - 7)(\left(- 12 x^{2} - 2 x - 9\right) \left(9 x^{3} + 2 x^{2} - 5 x - 7\right) + \left(27 x^{2} + 4 x - 5\right) \left(- 4 x^{3} - x^{2} - 9 x + 7\right))+(\left(- 4 x^{3} - x^{2} - 9 x + 7\right) \left(9 x^{3} + 2 x^{2} - 5 x - 7\right))(24 x^{2} - 4 x + 4)=\left(- 12 x^{2} - 2 x - 9\right) \left(8 x^{3} - 2 x^{2} + 4 x - 7\right) \left(9 x^{3} + 2 x^{2} - 5 x - 7\right) + \left(24 x^{2} - 4 x + 4\right) \left(- 4 x^{3} - x^{2} - 9 x + 7\right) \left(9 x^{3} + 2 x^{2} - 5 x - 7\right) + \left(27 x^{2} + 4 x - 5\right) \left(- 4 x^{3} - x^{2} - 9 x + 7\right) \left(8 x^{3} - 2 x^{2} + 4 x - 7\right) " src="/equation_images/%20%5Cdisplaystyle%20f%27%3D%288%20x%5E%7B3%7D%20-%202%20x%5E%7B2%7D%20%2B%204%20x%20-%207%29%28%5Cleft%28-%2012%20x%5E%7B2%7D%20-%202%20x%20-%209%5Cright%29%20%5Cleft%289%20x%5E%7B3%7D%20%2B%202%20x%5E%7B2%7D%20-%205%20x%20-%207%5Cright%29%20%2B%20%5Cleft%2827%20x%5E%7B2%7D%20%2B%204%20x%20-%205%5Cright%29%20%5Cleft%28-%204%20x%5E%7B3%7D%20-%20x%5E%7B2%7D%20-%209%20x%20%2B%207%5Cright%29%29%2B%28%5Cleft%28-%204%20x%5E%7B3%7D%20-%20x%5E%7B2%7D%20-%209%20x%20%2B%207%5Cright%29%20%5Cleft%289%20x%5E%7B3%7D%20%2B%202%20x%5E%7B2%7D%20-%205%20x%20-%207%5Cright%29%29%2824%20x%5E%7B2%7D%20-%204%20x%20%2B%204%29%3D%5Cleft%28-%2012%20x%5E%7B2%7D%20-%202%20x%20-%209%5Cright%29%20%5Cleft%288%20x%5E%7B3%7D%20-%202%20x%5E%7B2%7D%20%2B%204%20x%20-%207%5Cright%29%20%5Cleft%289%20x%5E%7B3%7D%20%2B%202%20x%5E%7B2%7D%20-%205%20x%20-%207%5Cright%29%20%2B%20%5Cleft%2824%20x%5E%7B2%7D%20-%204%20x%20%2B%204%5Cright%29%20%5Cleft%28-%204%20x%5E%7B3%7D%20-%20x%5E%7B2%7D%20-%209%20x%20%2B%207%5Cright%29%20%5Cleft%289%20x%5E%7B3%7D%20%2B%202%20x%5E%7B2%7D%20-%205%20x%20-%207%5Cright%29%20%2B%20%5Cleft%2827%20x%5E%7B2%7D%20%2B%204%20x%20-%205%5Cright%29%20%5Cleft%28-%204%20x%5E%7B3%7D%20-%20x%5E%7B2%7D%20-%209%20x%20%2B%207%5Cright%29%20%5Cleft%288%20x%5E%7B3%7D%20-%202%20x%5E%7B2%7D%20%2B%204%20x%20-%207%5Cright%29%20" alt="LaTeX: \displaystyle f'=(8 x^{3} - 2 x^{2} + 4 x - 7)(\left(- 12 x^{2} - 2 x - 9\right) \left(9 x^{3} + 2 x^{2} - 5 x - 7\right) + \left(27 x^{2} + 4 x - 5\right) \left(- 4 x^{3} - x^{2} - 9 x + 7\right))+(\left(- 4 x^{3} - x^{2} - 9 x + 7\right) \left(9 x^{3} + 2 x^{2} - 5 x - 7\right))(24 x^{2} - 4 x + 4)=\left(- 12 x^{2} - 2 x - 9\right) \left(8 x^{3} - 2 x^{2} + 4 x - 7\right) \left(9 x^{3} + 2 x^{2} - 5 x - 7\right) + \left(24 x^{2} - 4 x + 4\right) \left(- 4 x^{3} - x^{2} - 9 x + 7\right) \left(9 x^{3} + 2 x^{2} - 5 x - 7\right) + \left(27 x^{2} + 4 x - 5\right) \left(- 4 x^{3} - x^{2} - 9 x + 7\right) \left(8 x^{3} - 2 x^{2} + 4 x - 7\right) " data-equation-content=" \displaystyle f'=(8 x^{3} - 2 x^{2} + 4 x - 7)(\left(- 12 x^{2} - 2 x - 9\right) \left(9 x^{3} + 2 x^{2} - 5 x - 7\right) + \left(27 x^{2} + 4 x - 5\right) \left(- 4 x^{3} - x^{2} - 9 x + 7\right))+(\left(- 4 x^{3} - x^{2} - 9 x + 7\right) \left(9 x^{3} + 2 x^{2} - 5 x - 7\right))(24 x^{2} - 4 x + 4)=\left(- 12 x^{2} - 2 x - 9\right) \left(8 x^{3} - 2 x^{2} + 4 x - 7\right) \left(9 x^{3} + 2 x^{2} - 5 x - 7\right) + \left(24 x^{2} - 4 x + 4\right) \left(- 4 x^{3} - x^{2} - 9 x + 7\right) \left(9 x^{3} + 2 x^{2} - 5 x - 7\right) + \left(27 x^{2} + 4 x - 5\right) \left(- 4 x^{3} - x^{2} - 9 x + 7\right) \left(8 x^{3} - 2 x^{2} + 4 x - 7\right) " /> </p> </p>