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Find the derivative of \(\displaystyle y = (- 6 x^{3} - 3 x^{2} + 3 x - 6)(5 x^{3} + 3 x^{2} + 5 x + 2)(6 x^{3} + x^{2} - 7 x + 8)\).
Identifying \(\displaystyle f=- 6 x^{3} - 3 x^{2} + 3 x - 6\) and \(\displaystyle g=\left(5 x^{3} + 3 x^{2} + 5 x + 2\right) \left(6 x^{3} + x^{2} - 7 x + 8\right)\) and using the product rule with \(\displaystyle f=- 6 x^{3} - 3 x^{2} + 3 x - 6 \implies f'=- 18 x^{2} - 6 x + 3\). This leaves g as \(\displaystyle g = \left(5 x^{3} + 3 x^{2} + 5 x + 2\right) \left(6 x^{3} + x^{2} - 7 x + 8\right)\) which also requires the product rule. Pushing down in the new product rule \(\displaystyle f=5 x^{3} + 3 x^{2} + 5 x + 2 \implies f'=15 x^{2} + 6 x + 5\) and \(\displaystyle g=6 x^{3} + x^{2} - 7 x + 8 \implies g'=18 x^{2} + 2 x - 7\). Popping up a level gives \(\displaystyle g'=(6 x^{3} + x^{2} - 7 x + 8)(15 x^{2} + 6 x + 5)+(5 x^{3} + 3 x^{2} + 5 x + 2)(18 x^{2} + 2 x - 7)\)Popping up again (Back to the original problem) gives \(\displaystyle f'=(- 6 x^{3} - 3 x^{2} + 3 x - 6)(\left(15 x^{2} + 6 x + 5\right) \left(6 x^{3} + x^{2} - 7 x + 8\right) + \left(18 x^{2} + 2 x - 7\right) \left(5 x^{3} + 3 x^{2} + 5 x + 2\right))+(\left(5 x^{3} + 3 x^{2} + 5 x + 2\right) \left(6 x^{3} + x^{2} - 7 x + 8\right))(- 18 x^{2} - 6 x + 3)=\left(- 18 x^{2} - 6 x + 3\right) \left(5 x^{3} + 3 x^{2} + 5 x + 2\right) \left(6 x^{3} + x^{2} - 7 x + 8\right) + \left(15 x^{2} + 6 x + 5\right) \left(- 6 x^{3} - 3 x^{2} + 3 x - 6\right) \left(6 x^{3} + x^{2} - 7 x + 8\right) + \left(18 x^{2} + 2 x - 7\right) \left(- 6 x^{3} - 3 x^{2} + 3 x - 6\right) \left(5 x^{3} + 3 x^{2} + 5 x + 2\right)\)
\begin{question}Find the derivative of $y = (- 6 x^{3} - 3 x^{2} + 3 x - 6)(5 x^{3} + 3 x^{2} + 5 x + 2)(6 x^{3} + x^{2} - 7 x + 8)$.
\soln{9cm}{Identifying $f=- 6 x^{3} - 3 x^{2} + 3 x - 6$ and $g=\left(5 x^{3} + 3 x^{2} + 5 x + 2\right) \left(6 x^{3} + x^{2} - 7 x + 8\right)$ and using the product rule with $f=- 6 x^{3} - 3 x^{2} + 3 x - 6 \implies f'=- 18 x^{2} - 6 x + 3$. This leaves g as $g = \left(5 x^{3} + 3 x^{2} + 5 x + 2\right) \left(6 x^{3} + x^{2} - 7 x + 8\right)$ which also requires the product rule. Pushing down in the new product rule $f=5 x^{3} + 3 x^{2} + 5 x + 2 \implies f'=15 x^{2} + 6 x + 5$ and $g=6 x^{3} + x^{2} - 7 x + 8 \implies g'=18 x^{2} + 2 x - 7$. Popping up a level gives $g'=(6 x^{3} + x^{2} - 7 x + 8)(15 x^{2} + 6 x + 5)+(5 x^{3} + 3 x^{2} + 5 x + 2)(18 x^{2} + 2 x - 7)$Popping up again (Back to the original problem) gives $f'=(- 6 x^{3} - 3 x^{2} + 3 x - 6)(\left(15 x^{2} + 6 x + 5\right) \left(6 x^{3} + x^{2} - 7 x + 8\right) + \left(18 x^{2} + 2 x - 7\right) \left(5 x^{3} + 3 x^{2} + 5 x + 2\right))+(\left(5 x^{3} + 3 x^{2} + 5 x + 2\right) \left(6 x^{3} + x^{2} - 7 x + 8\right))(- 18 x^{2} - 6 x + 3)=\left(- 18 x^{2} - 6 x + 3\right) \left(5 x^{3} + 3 x^{2} + 5 x + 2\right) \left(6 x^{3} + x^{2} - 7 x + 8\right) + \left(15 x^{2} + 6 x + 5\right) \left(- 6 x^{3} - 3 x^{2} + 3 x - 6\right) \left(6 x^{3} + x^{2} - 7 x + 8\right) + \left(18 x^{2} + 2 x - 7\right) \left(- 6 x^{3} - 3 x^{2} + 3 x - 6\right) \left(5 x^{3} + 3 x^{2} + 5 x + 2\right)$}
\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 = (- 6 x^{3} - 3 x^{2} + 3 x - 6)(5 x^{3} + 3 x^{2} + 5 x + 2)(6 x^{3} + x^{2} - 7 x + 8) " src="/equation_images/%20%5Cdisplaystyle%20y%20%3D%20%28-%206%20x%5E%7B3%7D%20-%203%20x%5E%7B2%7D%20%2B%203%20x%20-%206%29%285%20x%5E%7B3%7D%20%2B%203%20x%5E%7B2%7D%20%2B%205%20x%20%2B%202%29%286%20x%5E%7B3%7D%20%2B%20x%5E%7B2%7D%20-%207%20x%20%2B%208%29%20" alt="LaTeX: \displaystyle y = (- 6 x^{3} - 3 x^{2} + 3 x - 6)(5 x^{3} + 3 x^{2} + 5 x + 2)(6 x^{3} + x^{2} - 7 x + 8) " data-equation-content=" \displaystyle y = (- 6 x^{3} - 3 x^{2} + 3 x - 6)(5 x^{3} + 3 x^{2} + 5 x + 2)(6 x^{3} + x^{2} - 7 x + 8) " /> .</p> </p><p> <p>Identifying <img class="equation_image" title=" \displaystyle f=- 6 x^{3} - 3 x^{2} + 3 x - 6 " src="/equation_images/%20%5Cdisplaystyle%20f%3D-%206%20x%5E%7B3%7D%20-%203%20x%5E%7B2%7D%20%2B%203%20x%20-%206%20" alt="LaTeX: \displaystyle f=- 6 x^{3} - 3 x^{2} + 3 x - 6 " data-equation-content=" \displaystyle f=- 6 x^{3} - 3 x^{2} + 3 x - 6 " /> and <img class="equation_image" title=" \displaystyle g=\left(5 x^{3} + 3 x^{2} + 5 x + 2\right) \left(6 x^{3} + x^{2} - 7 x + 8\right) " src="/equation_images/%20%5Cdisplaystyle%20g%3D%5Cleft%285%20x%5E%7B3%7D%20%2B%203%20x%5E%7B2%7D%20%2B%205%20x%20%2B%202%5Cright%29%20%5Cleft%286%20x%5E%7B3%7D%20%2B%20x%5E%7B2%7D%20-%207%20x%20%2B%208%5Cright%29%20" alt="LaTeX: \displaystyle g=\left(5 x^{3} + 3 x^{2} + 5 x + 2\right) \left(6 x^{3} + x^{2} - 7 x + 8\right) " data-equation-content=" \displaystyle g=\left(5 x^{3} + 3 x^{2} + 5 x + 2\right) \left(6 x^{3} + x^{2} - 7 x + 8\right) " /> and using the product rule with <img class="equation_image" title=" \displaystyle f=- 6 x^{3} - 3 x^{2} + 3 x - 6 \implies f'=- 18 x^{2} - 6 x + 3 " src="/equation_images/%20%5Cdisplaystyle%20f%3D-%206%20x%5E%7B3%7D%20-%203%20x%5E%7B2%7D%20%2B%203%20x%20-%206%20%5Cimplies%20f%27%3D-%2018%20x%5E%7B2%7D%20-%206%20x%20%2B%203%20" alt="LaTeX: \displaystyle f=- 6 x^{3} - 3 x^{2} + 3 x - 6 \implies f'=- 18 x^{2} - 6 x + 3 " data-equation-content=" \displaystyle f=- 6 x^{3} - 3 x^{2} + 3 x - 6 \implies f'=- 18 x^{2} - 6 x + 3 " /> . This leaves g as <img class="equation_image" title=" \displaystyle g = \left(5 x^{3} + 3 x^{2} + 5 x + 2\right) \left(6 x^{3} + x^{2} - 7 x + 8\right) " src="/equation_images/%20%5Cdisplaystyle%20g%20%3D%20%5Cleft%285%20x%5E%7B3%7D%20%2B%203%20x%5E%7B2%7D%20%2B%205%20x%20%2B%202%5Cright%29%20%5Cleft%286%20x%5E%7B3%7D%20%2B%20x%5E%7B2%7D%20-%207%20x%20%2B%208%5Cright%29%20" alt="LaTeX: \displaystyle g = \left(5 x^{3} + 3 x^{2} + 5 x + 2\right) \left(6 x^{3} + x^{2} - 7 x + 8\right) " data-equation-content=" \displaystyle g = \left(5 x^{3} + 3 x^{2} + 5 x + 2\right) \left(6 x^{3} + x^{2} - 7 x + 8\right) " /> which also requires the product rule. Pushing down in the new product rule <img class="equation_image" title=" \displaystyle f=5 x^{3} + 3 x^{2} + 5 x + 2 \implies f'=15 x^{2} + 6 x + 5 " src="/equation_images/%20%5Cdisplaystyle%20f%3D5%20x%5E%7B3%7D%20%2B%203%20x%5E%7B2%7D%20%2B%205%20x%20%2B%202%20%5Cimplies%20f%27%3D15%20x%5E%7B2%7D%20%2B%206%20x%20%2B%205%20" alt="LaTeX: \displaystyle f=5 x^{3} + 3 x^{2} + 5 x + 2 \implies f'=15 x^{2} + 6 x + 5 " data-equation-content=" \displaystyle f=5 x^{3} + 3 x^{2} + 5 x + 2 \implies f'=15 x^{2} + 6 x + 5 " /> and <img class="equation_image" title=" \displaystyle g=6 x^{3} + x^{2} - 7 x + 8 \implies g'=18 x^{2} + 2 x - 7 " src="/equation_images/%20%5Cdisplaystyle%20g%3D6%20x%5E%7B3%7D%20%2B%20x%5E%7B2%7D%20-%207%20x%20%2B%208%20%5Cimplies%20g%27%3D18%20x%5E%7B2%7D%20%2B%202%20x%20-%207%20" alt="LaTeX: \displaystyle g=6 x^{3} + x^{2} - 7 x + 8 \implies g'=18 x^{2} + 2 x - 7 " data-equation-content=" \displaystyle g=6 x^{3} + x^{2} - 7 x + 8 \implies g'=18 x^{2} + 2 x - 7 " /> . Popping up a level gives <img class="equation_image" title=" \displaystyle g'=(6 x^{3} + x^{2} - 7 x + 8)(15 x^{2} + 6 x + 5)+(5 x^{3} + 3 x^{2} + 5 x + 2)(18 x^{2} + 2 x - 7) " src="/equation_images/%20%5Cdisplaystyle%20g%27%3D%286%20x%5E%7B3%7D%20%2B%20x%5E%7B2%7D%20-%207%20x%20%2B%208%29%2815%20x%5E%7B2%7D%20%2B%206%20x%20%2B%205%29%2B%285%20x%5E%7B3%7D%20%2B%203%20x%5E%7B2%7D%20%2B%205%20x%20%2B%202%29%2818%20x%5E%7B2%7D%20%2B%202%20x%20-%207%29%20" alt="LaTeX: \displaystyle g'=(6 x^{3} + x^{2} - 7 x + 8)(15 x^{2} + 6 x + 5)+(5 x^{3} + 3 x^{2} + 5 x + 2)(18 x^{2} + 2 x - 7) " data-equation-content=" \displaystyle g'=(6 x^{3} + x^{2} - 7 x + 8)(15 x^{2} + 6 x + 5)+(5 x^{3} + 3 x^{2} + 5 x + 2)(18 x^{2} + 2 x - 7) " /> Popping up again (Back to the original problem) gives <img class="equation_image" title=" \displaystyle f'=(- 6 x^{3} - 3 x^{2} + 3 x - 6)(\left(15 x^{2} + 6 x + 5\right) \left(6 x^{3} + x^{2} - 7 x + 8\right) + \left(18 x^{2} + 2 x - 7\right) \left(5 x^{3} + 3 x^{2} + 5 x + 2\right))+(\left(5 x^{3} + 3 x^{2} + 5 x + 2\right) \left(6 x^{3} + x^{2} - 7 x + 8\right))(- 18 x^{2} - 6 x + 3)=\left(- 18 x^{2} - 6 x + 3\right) \left(5 x^{3} + 3 x^{2} + 5 x + 2\right) \left(6 x^{3} + x^{2} - 7 x + 8\right) + \left(15 x^{2} + 6 x + 5\right) \left(- 6 x^{3} - 3 x^{2} + 3 x - 6\right) \left(6 x^{3} + x^{2} - 7 x + 8\right) + \left(18 x^{2} + 2 x - 7\right) \left(- 6 x^{3} - 3 x^{2} + 3 x - 6\right) \left(5 x^{3} + 3 x^{2} + 5 x + 2\right) " src="/equation_images/%20%5Cdisplaystyle%20f%27%3D%28-%206%20x%5E%7B3%7D%20-%203%20x%5E%7B2%7D%20%2B%203%20x%20-%206%29%28%5Cleft%2815%20x%5E%7B2%7D%20%2B%206%20x%20%2B%205%5Cright%29%20%5Cleft%286%20x%5E%7B3%7D%20%2B%20x%5E%7B2%7D%20-%207%20x%20%2B%208%5Cright%29%20%2B%20%5Cleft%2818%20x%5E%7B2%7D%20%2B%202%20x%20-%207%5Cright%29%20%5Cleft%285%20x%5E%7B3%7D%20%2B%203%20x%5E%7B2%7D%20%2B%205%20x%20%2B%202%5Cright%29%29%2B%28%5Cleft%285%20x%5E%7B3%7D%20%2B%203%20x%5E%7B2%7D%20%2B%205%20x%20%2B%202%5Cright%29%20%5Cleft%286%20x%5E%7B3%7D%20%2B%20x%5E%7B2%7D%20-%207%20x%20%2B%208%5Cright%29%29%28-%2018%20x%5E%7B2%7D%20-%206%20x%20%2B%203%29%3D%5Cleft%28-%2018%20x%5E%7B2%7D%20-%206%20x%20%2B%203%5Cright%29%20%5Cleft%285%20x%5E%7B3%7D%20%2B%203%20x%5E%7B2%7D%20%2B%205%20x%20%2B%202%5Cright%29%20%5Cleft%286%20x%5E%7B3%7D%20%2B%20x%5E%7B2%7D%20-%207%20x%20%2B%208%5Cright%29%20%2B%20%5Cleft%2815%20x%5E%7B2%7D%20%2B%206%20x%20%2B%205%5Cright%29%20%5Cleft%28-%206%20x%5E%7B3%7D%20-%203%20x%5E%7B2%7D%20%2B%203%20x%20-%206%5Cright%29%20%5Cleft%286%20x%5E%7B3%7D%20%2B%20x%5E%7B2%7D%20-%207%20x%20%2B%208%5Cright%29%20%2B%20%5Cleft%2818%20x%5E%7B2%7D%20%2B%202%20x%20-%207%5Cright%29%20%5Cleft%28-%206%20x%5E%7B3%7D%20-%203%20x%5E%7B2%7D%20%2B%203%20x%20-%206%5Cright%29%20%5Cleft%285%20x%5E%7B3%7D%20%2B%203%20x%5E%7B2%7D%20%2B%205%20x%20%2B%202%5Cright%29%20" alt="LaTeX: \displaystyle f'=(- 6 x^{3} - 3 x^{2} + 3 x - 6)(\left(15 x^{2} + 6 x + 5\right) \left(6 x^{3} + x^{2} - 7 x + 8\right) + \left(18 x^{2} + 2 x - 7\right) \left(5 x^{3} + 3 x^{2} + 5 x + 2\right))+(\left(5 x^{3} + 3 x^{2} + 5 x + 2\right) \left(6 x^{3} + x^{2} - 7 x + 8\right))(- 18 x^{2} - 6 x + 3)=\left(- 18 x^{2} - 6 x + 3\right) \left(5 x^{3} + 3 x^{2} + 5 x + 2\right) \left(6 x^{3} + x^{2} - 7 x + 8\right) + \left(15 x^{2} + 6 x + 5\right) \left(- 6 x^{3} - 3 x^{2} + 3 x - 6\right) \left(6 x^{3} + x^{2} - 7 x + 8\right) + \left(18 x^{2} + 2 x - 7\right) \left(- 6 x^{3} - 3 x^{2} + 3 x - 6\right) \left(5 x^{3} + 3 x^{2} + 5 x + 2\right) " data-equation-content=" \displaystyle f'=(- 6 x^{3} - 3 x^{2} + 3 x - 6)(\left(15 x^{2} + 6 x + 5\right) \left(6 x^{3} + x^{2} - 7 x + 8\right) + \left(18 x^{2} + 2 x - 7\right) \left(5 x^{3} + 3 x^{2} + 5 x + 2\right))+(\left(5 x^{3} + 3 x^{2} + 5 x + 2\right) \left(6 x^{3} + x^{2} - 7 x + 8\right))(- 18 x^{2} - 6 x + 3)=\left(- 18 x^{2} - 6 x + 3\right) \left(5 x^{3} + 3 x^{2} + 5 x + 2\right) \left(6 x^{3} + x^{2} - 7 x + 8\right) + \left(15 x^{2} + 6 x + 5\right) \left(- 6 x^{3} - 3 x^{2} + 3 x - 6\right) \left(6 x^{3} + x^{2} - 7 x + 8\right) + \left(18 x^{2} + 2 x - 7\right) \left(- 6 x^{3} - 3 x^{2} + 3 x - 6\right) \left(5 x^{3} + 3 x^{2} + 5 x + 2\right) " /> </p> </p>