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Find the derivative of \(\displaystyle y = (- 6 x^{3} - 3 x^{2} + 2 x - 7)(- 3 x^{3} + 5 x^{2} + 9 x + 6)(- 9 x^{3} - x^{2} + 5 x - 4)\).
Identifying \(\displaystyle f=- 6 x^{3} - 3 x^{2} + 2 x - 7\) and \(\displaystyle g=\left(- 9 x^{3} - x^{2} + 5 x - 4\right) \left(- 3 x^{3} + 5 x^{2} + 9 x + 6\right)\) and using the product rule with \(\displaystyle f=- 6 x^{3} - 3 x^{2} + 2 x - 7 \implies f'=- 18 x^{2} - 6 x + 2\). This leaves g as \(\displaystyle g = \left(- 9 x^{3} - x^{2} + 5 x - 4\right) \left(- 3 x^{3} + 5 x^{2} + 9 x + 6\right)\) which also requires the product rule. Pushing down in the new product rule \(\displaystyle f=- 3 x^{3} + 5 x^{2} + 9 x + 6 \implies f'=- 9 x^{2} + 10 x + 9\) and \(\displaystyle g=- 9 x^{3} - x^{2} + 5 x - 4 \implies g'=- 27 x^{2} - 2 x + 5\). Popping up a level gives \(\displaystyle g'=(- 9 x^{3} - x^{2} + 5 x - 4)(- 9 x^{2} + 10 x + 9)+(- 3 x^{3} + 5 x^{2} + 9 x + 6)(- 27 x^{2} - 2 x + 5)\)Popping up again (Back to the original problem) gives \(\displaystyle f'=(- 6 x^{3} - 3 x^{2} + 2 x - 7)(\left(- 27 x^{2} - 2 x + 5\right) \left(- 3 x^{3} + 5 x^{2} + 9 x + 6\right) + \left(- 9 x^{2} + 10 x + 9\right) \left(- 9 x^{3} - x^{2} + 5 x - 4\right))+(\left(- 9 x^{3} - x^{2} + 5 x - 4\right) \left(- 3 x^{3} + 5 x^{2} + 9 x + 6\right))(- 18 x^{2} - 6 x + 2)=\left(- 27 x^{2} - 2 x + 5\right) \left(- 6 x^{3} - 3 x^{2} + 2 x - 7\right) \left(- 3 x^{3} + 5 x^{2} + 9 x + 6\right) + \left(- 18 x^{2} - 6 x + 2\right) \left(- 9 x^{3} - x^{2} + 5 x - 4\right) \left(- 3 x^{3} + 5 x^{2} + 9 x + 6\right) + \left(- 9 x^{2} + 10 x + 9\right) \left(- 9 x^{3} - x^{2} + 5 x - 4\right) \left(- 6 x^{3} - 3 x^{2} + 2 x - 7\right)\)
\begin{question}Find the derivative of $y = (- 6 x^{3} - 3 x^{2} + 2 x - 7)(- 3 x^{3} + 5 x^{2} + 9 x + 6)(- 9 x^{3} - x^{2} + 5 x - 4)$.
\soln{9cm}{Identifying $f=- 6 x^{3} - 3 x^{2} + 2 x - 7$ and $g=\left(- 9 x^{3} - x^{2} + 5 x - 4\right) \left(- 3 x^{3} + 5 x^{2} + 9 x + 6\right)$ and using the product rule with $f=- 6 x^{3} - 3 x^{2} + 2 x - 7 \implies f'=- 18 x^{2} - 6 x + 2$. This leaves g as $g = \left(- 9 x^{3} - x^{2} + 5 x - 4\right) \left(- 3 x^{3} + 5 x^{2} + 9 x + 6\right)$ which also requires the product rule. Pushing down in the new product rule $f=- 3 x^{3} + 5 x^{2} + 9 x + 6 \implies f'=- 9 x^{2} + 10 x + 9$ and $g=- 9 x^{3} - x^{2} + 5 x - 4 \implies g'=- 27 x^{2} - 2 x + 5$. Popping up a level gives $g'=(- 9 x^{3} - x^{2} + 5 x - 4)(- 9 x^{2} + 10 x + 9)+(- 3 x^{3} + 5 x^{2} + 9 x + 6)(- 27 x^{2} - 2 x + 5)$Popping up again (Back to the original problem) gives $f'=(- 6 x^{3} - 3 x^{2} + 2 x - 7)(\left(- 27 x^{2} - 2 x + 5\right) \left(- 3 x^{3} + 5 x^{2} + 9 x + 6\right) + \left(- 9 x^{2} + 10 x + 9\right) \left(- 9 x^{3} - x^{2} + 5 x - 4\right))+(\left(- 9 x^{3} - x^{2} + 5 x - 4\right) \left(- 3 x^{3} + 5 x^{2} + 9 x + 6\right))(- 18 x^{2} - 6 x + 2)=\left(- 27 x^{2} - 2 x + 5\right) \left(- 6 x^{3} - 3 x^{2} + 2 x - 7\right) \left(- 3 x^{3} + 5 x^{2} + 9 x + 6\right) + \left(- 18 x^{2} - 6 x + 2\right) \left(- 9 x^{3} - x^{2} + 5 x - 4\right) \left(- 3 x^{3} + 5 x^{2} + 9 x + 6\right) + \left(- 9 x^{2} + 10 x + 9\right) \left(- 9 x^{3} - x^{2} + 5 x - 4\right) \left(- 6 x^{3} - 3 x^{2} + 2 x - 7\right)$}
\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 = (- 6 x^{3} - 3 x^{2} + 2 x - 7)(- 3 x^{3} + 5 x^{2} + 9 x + 6)(- 9 x^{3} - x^{2} + 5 x - 4) " src="/equation_images/%20%5Cdisplaystyle%20y%20%3D%20%28-%206%20x%5E%7B3%7D%20-%203%20x%5E%7B2%7D%20%2B%202%20x%20-%207%29%28-%203%20x%5E%7B3%7D%20%2B%205%20x%5E%7B2%7D%20%2B%209%20x%20%2B%206%29%28-%209%20x%5E%7B3%7D%20-%20x%5E%7B2%7D%20%2B%205%20x%20-%204%29%20" alt="LaTeX: \displaystyle y = (- 6 x^{3} - 3 x^{2} + 2 x - 7)(- 3 x^{3} + 5 x^{2} + 9 x + 6)(- 9 x^{3} - x^{2} + 5 x - 4) " data-equation-content=" \displaystyle y = (- 6 x^{3} - 3 x^{2} + 2 x - 7)(- 3 x^{3} + 5 x^{2} + 9 x + 6)(- 9 x^{3} - x^{2} + 5 x - 4) " /> .</p> </p><p> <p>Identifying <img class="equation_image" title=" \displaystyle f=- 6 x^{3} - 3 x^{2} + 2 x - 7 " src="/equation_images/%20%5Cdisplaystyle%20f%3D-%206%20x%5E%7B3%7D%20-%203%20x%5E%7B2%7D%20%2B%202%20x%20-%207%20" alt="LaTeX: \displaystyle f=- 6 x^{3} - 3 x^{2} + 2 x - 7 " data-equation-content=" \displaystyle f=- 6 x^{3} - 3 x^{2} + 2 x - 7 " /> and <img class="equation_image" title=" \displaystyle g=\left(- 9 x^{3} - x^{2} + 5 x - 4\right) \left(- 3 x^{3} + 5 x^{2} + 9 x + 6\right) " src="/equation_images/%20%5Cdisplaystyle%20g%3D%5Cleft%28-%209%20x%5E%7B3%7D%20-%20x%5E%7B2%7D%20%2B%205%20x%20-%204%5Cright%29%20%5Cleft%28-%203%20x%5E%7B3%7D%20%2B%205%20x%5E%7B2%7D%20%2B%209%20x%20%2B%206%5Cright%29%20" alt="LaTeX: \displaystyle g=\left(- 9 x^{3} - x^{2} + 5 x - 4\right) \left(- 3 x^{3} + 5 x^{2} + 9 x + 6\right) " data-equation-content=" \displaystyle g=\left(- 9 x^{3} - x^{2} + 5 x - 4\right) \left(- 3 x^{3} + 5 x^{2} + 9 x + 6\right) " /> and using the product rule with <img class="equation_image" title=" \displaystyle f=- 6 x^{3} - 3 x^{2} + 2 x - 7 \implies f'=- 18 x^{2} - 6 x + 2 " src="/equation_images/%20%5Cdisplaystyle%20f%3D-%206%20x%5E%7B3%7D%20-%203%20x%5E%7B2%7D%20%2B%202%20x%20-%207%20%5Cimplies%20f%27%3D-%2018%20x%5E%7B2%7D%20-%206%20x%20%2B%202%20" alt="LaTeX: \displaystyle f=- 6 x^{3} - 3 x^{2} + 2 x - 7 \implies f'=- 18 x^{2} - 6 x + 2 " data-equation-content=" \displaystyle f=- 6 x^{3} - 3 x^{2} + 2 x - 7 \implies f'=- 18 x^{2} - 6 x + 2 " /> . This leaves g as <img class="equation_image" title=" \displaystyle g = \left(- 9 x^{3} - x^{2} + 5 x - 4\right) \left(- 3 x^{3} + 5 x^{2} + 9 x + 6\right) " src="/equation_images/%20%5Cdisplaystyle%20g%20%3D%20%5Cleft%28-%209%20x%5E%7B3%7D%20-%20x%5E%7B2%7D%20%2B%205%20x%20-%204%5Cright%29%20%5Cleft%28-%203%20x%5E%7B3%7D%20%2B%205%20x%5E%7B2%7D%20%2B%209%20x%20%2B%206%5Cright%29%20" alt="LaTeX: \displaystyle g = \left(- 9 x^{3} - x^{2} + 5 x - 4\right) \left(- 3 x^{3} + 5 x^{2} + 9 x + 6\right) " data-equation-content=" \displaystyle g = \left(- 9 x^{3} - x^{2} + 5 x - 4\right) \left(- 3 x^{3} + 5 x^{2} + 9 x + 6\right) " /> which also requires the product rule. Pushing down in the new product rule <img class="equation_image" title=" \displaystyle f=- 3 x^{3} + 5 x^{2} + 9 x + 6 \implies f'=- 9 x^{2} + 10 x + 9 " src="/equation_images/%20%5Cdisplaystyle%20f%3D-%203%20x%5E%7B3%7D%20%2B%205%20x%5E%7B2%7D%20%2B%209%20x%20%2B%206%20%5Cimplies%20f%27%3D-%209%20x%5E%7B2%7D%20%2B%2010%20x%20%2B%209%20" alt="LaTeX: \displaystyle f=- 3 x^{3} + 5 x^{2} + 9 x + 6 \implies f'=- 9 x^{2} + 10 x + 9 " data-equation-content=" \displaystyle f=- 3 x^{3} + 5 x^{2} + 9 x + 6 \implies f'=- 9 x^{2} + 10 x + 9 " /> and <img class="equation_image" title=" \displaystyle g=- 9 x^{3} - x^{2} + 5 x - 4 \implies g'=- 27 x^{2} - 2 x + 5 " src="/equation_images/%20%5Cdisplaystyle%20g%3D-%209%20x%5E%7B3%7D%20-%20x%5E%7B2%7D%20%2B%205%20x%20-%204%20%5Cimplies%20g%27%3D-%2027%20x%5E%7B2%7D%20-%202%20x%20%2B%205%20" alt="LaTeX: \displaystyle g=- 9 x^{3} - x^{2} + 5 x - 4 \implies g'=- 27 x^{2} - 2 x + 5 " data-equation-content=" \displaystyle g=- 9 x^{3} - x^{2} + 5 x - 4 \implies g'=- 27 x^{2} - 2 x + 5 " /> . Popping up a level gives <img class="equation_image" title=" \displaystyle g'=(- 9 x^{3} - x^{2} + 5 x - 4)(- 9 x^{2} + 10 x + 9)+(- 3 x^{3} + 5 x^{2} + 9 x + 6)(- 27 x^{2} - 2 x + 5) " src="/equation_images/%20%5Cdisplaystyle%20g%27%3D%28-%209%20x%5E%7B3%7D%20-%20x%5E%7B2%7D%20%2B%205%20x%20-%204%29%28-%209%20x%5E%7B2%7D%20%2B%2010%20x%20%2B%209%29%2B%28-%203%20x%5E%7B3%7D%20%2B%205%20x%5E%7B2%7D%20%2B%209%20x%20%2B%206%29%28-%2027%20x%5E%7B2%7D%20-%202%20x%20%2B%205%29%20" alt="LaTeX: \displaystyle g'=(- 9 x^{3} - x^{2} + 5 x - 4)(- 9 x^{2} + 10 x + 9)+(- 3 x^{3} + 5 x^{2} + 9 x + 6)(- 27 x^{2} - 2 x + 5) " data-equation-content=" \displaystyle g'=(- 9 x^{3} - x^{2} + 5 x - 4)(- 9 x^{2} + 10 x + 9)+(- 3 x^{3} + 5 x^{2} + 9 x + 6)(- 27 x^{2} - 2 x + 5) " /> Popping up again (Back to the original problem) gives <img class="equation_image" title=" \displaystyle f'=(- 6 x^{3} - 3 x^{2} + 2 x - 7)(\left(- 27 x^{2} - 2 x + 5\right) \left(- 3 x^{3} + 5 x^{2} + 9 x + 6\right) + \left(- 9 x^{2} + 10 x + 9\right) \left(- 9 x^{3} - x^{2} + 5 x - 4\right))+(\left(- 9 x^{3} - x^{2} + 5 x - 4\right) \left(- 3 x^{3} + 5 x^{2} + 9 x + 6\right))(- 18 x^{2} - 6 x + 2)=\left(- 27 x^{2} - 2 x + 5\right) \left(- 6 x^{3} - 3 x^{2} + 2 x - 7\right) \left(- 3 x^{3} + 5 x^{2} + 9 x + 6\right) + \left(- 18 x^{2} - 6 x + 2\right) \left(- 9 x^{3} - x^{2} + 5 x - 4\right) \left(- 3 x^{3} + 5 x^{2} + 9 x + 6\right) + \left(- 9 x^{2} + 10 x + 9\right) \left(- 9 x^{3} - x^{2} + 5 x - 4\right) \left(- 6 x^{3} - 3 x^{2} + 2 x - 7\right) " src="/equation_images/%20%5Cdisplaystyle%20f%27%3D%28-%206%20x%5E%7B3%7D%20-%203%20x%5E%7B2%7D%20%2B%202%20x%20-%207%29%28%5Cleft%28-%2027%20x%5E%7B2%7D%20-%202%20x%20%2B%205%5Cright%29%20%5Cleft%28-%203%20x%5E%7B3%7D%20%2B%205%20x%5E%7B2%7D%20%2B%209%20x%20%2B%206%5Cright%29%20%2B%20%5Cleft%28-%209%20x%5E%7B2%7D%20%2B%2010%20x%20%2B%209%5Cright%29%20%5Cleft%28-%209%20x%5E%7B3%7D%20-%20x%5E%7B2%7D%20%2B%205%20x%20-%204%5Cright%29%29%2B%28%5Cleft%28-%209%20x%5E%7B3%7D%20-%20x%5E%7B2%7D%20%2B%205%20x%20-%204%5Cright%29%20%5Cleft%28-%203%20x%5E%7B3%7D%20%2B%205%20x%5E%7B2%7D%20%2B%209%20x%20%2B%206%5Cright%29%29%28-%2018%20x%5E%7B2%7D%20-%206%20x%20%2B%202%29%3D%5Cleft%28-%2027%20x%5E%7B2%7D%20-%202%20x%20%2B%205%5Cright%29%20%5Cleft%28-%206%20x%5E%7B3%7D%20-%203%20x%5E%7B2%7D%20%2B%202%20x%20-%207%5Cright%29%20%5Cleft%28-%203%20x%5E%7B3%7D%20%2B%205%20x%5E%7B2%7D%20%2B%209%20x%20%2B%206%5Cright%29%20%2B%20%5Cleft%28-%2018%20x%5E%7B2%7D%20-%206%20x%20%2B%202%5Cright%29%20%5Cleft%28-%209%20x%5E%7B3%7D%20-%20x%5E%7B2%7D%20%2B%205%20x%20-%204%5Cright%29%20%5Cleft%28-%203%20x%5E%7B3%7D%20%2B%205%20x%5E%7B2%7D%20%2B%209%20x%20%2B%206%5Cright%29%20%2B%20%5Cleft%28-%209%20x%5E%7B2%7D%20%2B%2010%20x%20%2B%209%5Cright%29%20%5Cleft%28-%209%20x%5E%7B3%7D%20-%20x%5E%7B2%7D%20%2B%205%20x%20-%204%5Cright%29%20%5Cleft%28-%206%20x%5E%7B3%7D%20-%203%20x%5E%7B2%7D%20%2B%202%20x%20-%207%5Cright%29%20" alt="LaTeX: \displaystyle f'=(- 6 x^{3} - 3 x^{2} + 2 x - 7)(\left(- 27 x^{2} - 2 x + 5\right) \left(- 3 x^{3} + 5 x^{2} + 9 x + 6\right) + \left(- 9 x^{2} + 10 x + 9\right) \left(- 9 x^{3} - x^{2} + 5 x - 4\right))+(\left(- 9 x^{3} - x^{2} + 5 x - 4\right) \left(- 3 x^{3} + 5 x^{2} + 9 x + 6\right))(- 18 x^{2} - 6 x + 2)=\left(- 27 x^{2} - 2 x + 5\right) \left(- 6 x^{3} - 3 x^{2} + 2 x - 7\right) \left(- 3 x^{3} + 5 x^{2} + 9 x + 6\right) + \left(- 18 x^{2} - 6 x + 2\right) \left(- 9 x^{3} - x^{2} + 5 x - 4\right) \left(- 3 x^{3} + 5 x^{2} + 9 x + 6\right) + \left(- 9 x^{2} + 10 x + 9\right) \left(- 9 x^{3} - x^{2} + 5 x - 4\right) \left(- 6 x^{3} - 3 x^{2} + 2 x - 7\right) " data-equation-content=" \displaystyle f'=(- 6 x^{3} - 3 x^{2} + 2 x - 7)(\left(- 27 x^{2} - 2 x + 5\right) \left(- 3 x^{3} + 5 x^{2} + 9 x + 6\right) + \left(- 9 x^{2} + 10 x + 9\right) \left(- 9 x^{3} - x^{2} + 5 x - 4\right))+(\left(- 9 x^{3} - x^{2} + 5 x - 4\right) \left(- 3 x^{3} + 5 x^{2} + 9 x + 6\right))(- 18 x^{2} - 6 x + 2)=\left(- 27 x^{2} - 2 x + 5\right) \left(- 6 x^{3} - 3 x^{2} + 2 x - 7\right) \left(- 3 x^{3} + 5 x^{2} + 9 x + 6\right) + \left(- 18 x^{2} - 6 x + 2\right) \left(- 9 x^{3} - x^{2} + 5 x - 4\right) \left(- 3 x^{3} + 5 x^{2} + 9 x + 6\right) + \left(- 9 x^{2} + 10 x + 9\right) \left(- 9 x^{3} - x^{2} + 5 x - 4\right) \left(- 6 x^{3} - 3 x^{2} + 2 x - 7\right) " /> </p> </p>