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Find the derivative of \(\displaystyle y = (- 2 x^{3} + 7 x^{2} + 6 x - 5)(- 6 x^{3} - 4 x^{2} + 7 x - 4)(- 7 x^{3} - 5 x^{2} + 7 x - 6)\).
Identifying \(\displaystyle f=- 2 x^{3} + 7 x^{2} + 6 x - 5\) and \(\displaystyle g=\left(- 7 x^{3} - 5 x^{2} + 7 x - 6\right) \left(- 6 x^{3} - 4 x^{2} + 7 x - 4\right)\) and using the product rule with \(\displaystyle f=- 2 x^{3} + 7 x^{2} + 6 x - 5 \implies f'=- 6 x^{2} + 14 x + 6\). This leaves g as \(\displaystyle g = \left(- 7 x^{3} - 5 x^{2} + 7 x - 6\right) \left(- 6 x^{3} - 4 x^{2} + 7 x - 4\right)\) which also requires the product rule. Pushing down in the new product rule \(\displaystyle f=- 6 x^{3} - 4 x^{2} + 7 x - 4 \implies f'=- 18 x^{2} - 8 x + 7\) and \(\displaystyle g=- 7 x^{3} - 5 x^{2} + 7 x - 6 \implies g'=- 21 x^{2} - 10 x + 7\). Popping up a level gives \(\displaystyle g'=(- 7 x^{3} - 5 x^{2} + 7 x - 6)(- 18 x^{2} - 8 x + 7)+(- 6 x^{3} - 4 x^{2} + 7 x - 4)(- 21 x^{2} - 10 x + 7)\)Popping up again (Back to the original problem) gives \(\displaystyle f'=(- 2 x^{3} + 7 x^{2} + 6 x - 5)(\left(- 21 x^{2} - 10 x + 7\right) \left(- 6 x^{3} - 4 x^{2} + 7 x - 4\right) + \left(- 18 x^{2} - 8 x + 7\right) \left(- 7 x^{3} - 5 x^{2} + 7 x - 6\right))+(\left(- 7 x^{3} - 5 x^{2} + 7 x - 6\right) \left(- 6 x^{3} - 4 x^{2} + 7 x - 4\right))(- 6 x^{2} + 14 x + 6)=\left(- 21 x^{2} - 10 x + 7\right) \left(- 6 x^{3} - 4 x^{2} + 7 x - 4\right) \left(- 2 x^{3} + 7 x^{2} + 6 x - 5\right) + \left(- 18 x^{2} - 8 x + 7\right) \left(- 7 x^{3} - 5 x^{2} + 7 x - 6\right) \left(- 2 x^{3} + 7 x^{2} + 6 x - 5\right) + \left(- 6 x^{2} + 14 x + 6\right) \left(- 7 x^{3} - 5 x^{2} + 7 x - 6\right) \left(- 6 x^{3} - 4 x^{2} + 7 x - 4\right)\)
\begin{question}Find the derivative of $y = (- 2 x^{3} + 7 x^{2} + 6 x - 5)(- 6 x^{3} - 4 x^{2} + 7 x - 4)(- 7 x^{3} - 5 x^{2} + 7 x - 6)$.
\soln{9cm}{Identifying $f=- 2 x^{3} + 7 x^{2} + 6 x - 5$ and $g=\left(- 7 x^{3} - 5 x^{2} + 7 x - 6\right) \left(- 6 x^{3} - 4 x^{2} + 7 x - 4\right)$ and using the product rule with $f=- 2 x^{3} + 7 x^{2} + 6 x - 5 \implies f'=- 6 x^{2} + 14 x + 6$. This leaves g as $g = \left(- 7 x^{3} - 5 x^{2} + 7 x - 6\right) \left(- 6 x^{3} - 4 x^{2} + 7 x - 4\right)$ which also requires the product rule. Pushing down in the new product rule $f=- 6 x^{3} - 4 x^{2} + 7 x - 4 \implies f'=- 18 x^{2} - 8 x + 7$ and $g=- 7 x^{3} - 5 x^{2} + 7 x - 6 \implies g'=- 21 x^{2} - 10 x + 7$. Popping up a level gives $g'=(- 7 x^{3} - 5 x^{2} + 7 x - 6)(- 18 x^{2} - 8 x + 7)+(- 6 x^{3} - 4 x^{2} + 7 x - 4)(- 21 x^{2} - 10 x + 7)$Popping up again (Back to the original problem) gives $f'=(- 2 x^{3} + 7 x^{2} + 6 x - 5)(\left(- 21 x^{2} - 10 x + 7\right) \left(- 6 x^{3} - 4 x^{2} + 7 x - 4\right) + \left(- 18 x^{2} - 8 x + 7\right) \left(- 7 x^{3} - 5 x^{2} + 7 x - 6\right))+(\left(- 7 x^{3} - 5 x^{2} + 7 x - 6\right) \left(- 6 x^{3} - 4 x^{2} + 7 x - 4\right))(- 6 x^{2} + 14 x + 6)=\left(- 21 x^{2} - 10 x + 7\right) \left(- 6 x^{3} - 4 x^{2} + 7 x - 4\right) \left(- 2 x^{3} + 7 x^{2} + 6 x - 5\right) + \left(- 18 x^{2} - 8 x + 7\right) \left(- 7 x^{3} - 5 x^{2} + 7 x - 6\right) \left(- 2 x^{3} + 7 x^{2} + 6 x - 5\right) + \left(- 6 x^{2} + 14 x + 6\right) \left(- 7 x^{3} - 5 x^{2} + 7 x - 6\right) \left(- 6 x^{3} - 4 x^{2} + 7 x - 4\right)$}
\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 = (- 2 x^{3} + 7 x^{2} + 6 x - 5)(- 6 x^{3} - 4 x^{2} + 7 x - 4)(- 7 x^{3} - 5 x^{2} + 7 x - 6) " src="/equation_images/%20%5Cdisplaystyle%20y%20%3D%20%28-%202%20x%5E%7B3%7D%20%2B%207%20x%5E%7B2%7D%20%2B%206%20x%20-%205%29%28-%206%20x%5E%7B3%7D%20-%204%20x%5E%7B2%7D%20%2B%207%20x%20-%204%29%28-%207%20x%5E%7B3%7D%20-%205%20x%5E%7B2%7D%20%2B%207%20x%20-%206%29%20" alt="LaTeX: \displaystyle y = (- 2 x^{3} + 7 x^{2} + 6 x - 5)(- 6 x^{3} - 4 x^{2} + 7 x - 4)(- 7 x^{3} - 5 x^{2} + 7 x - 6) " data-equation-content=" \displaystyle y = (- 2 x^{3} + 7 x^{2} + 6 x - 5)(- 6 x^{3} - 4 x^{2} + 7 x - 4)(- 7 x^{3} - 5 x^{2} + 7 x - 6) " /> .</p> </p><p> <p>Identifying <img class="equation_image" title=" \displaystyle f=- 2 x^{3} + 7 x^{2} + 6 x - 5 " src="/equation_images/%20%5Cdisplaystyle%20f%3D-%202%20x%5E%7B3%7D%20%2B%207%20x%5E%7B2%7D%20%2B%206%20x%20-%205%20" alt="LaTeX: \displaystyle f=- 2 x^{3} + 7 x^{2} + 6 x - 5 " data-equation-content=" \displaystyle f=- 2 x^{3} + 7 x^{2} + 6 x - 5 " /> and <img class="equation_image" title=" \displaystyle g=\left(- 7 x^{3} - 5 x^{2} + 7 x - 6\right) \left(- 6 x^{3} - 4 x^{2} + 7 x - 4\right) " src="/equation_images/%20%5Cdisplaystyle%20g%3D%5Cleft%28-%207%20x%5E%7B3%7D%20-%205%20x%5E%7B2%7D%20%2B%207%20x%20-%206%5Cright%29%20%5Cleft%28-%206%20x%5E%7B3%7D%20-%204%20x%5E%7B2%7D%20%2B%207%20x%20-%204%5Cright%29%20" alt="LaTeX: \displaystyle g=\left(- 7 x^{3} - 5 x^{2} + 7 x - 6\right) \left(- 6 x^{3} - 4 x^{2} + 7 x - 4\right) " data-equation-content=" \displaystyle g=\left(- 7 x^{3} - 5 x^{2} + 7 x - 6\right) \left(- 6 x^{3} - 4 x^{2} + 7 x - 4\right) " /> and using the product rule with <img class="equation_image" title=" \displaystyle f=- 2 x^{3} + 7 x^{2} + 6 x - 5 \implies f'=- 6 x^{2} + 14 x + 6 " src="/equation_images/%20%5Cdisplaystyle%20f%3D-%202%20x%5E%7B3%7D%20%2B%207%20x%5E%7B2%7D%20%2B%206%20x%20-%205%20%5Cimplies%20f%27%3D-%206%20x%5E%7B2%7D%20%2B%2014%20x%20%2B%206%20" alt="LaTeX: \displaystyle f=- 2 x^{3} + 7 x^{2} + 6 x - 5 \implies f'=- 6 x^{2} + 14 x + 6 " data-equation-content=" \displaystyle f=- 2 x^{3} + 7 x^{2} + 6 x - 5 \implies f'=- 6 x^{2} + 14 x + 6 " /> . This leaves g as <img class="equation_image" title=" \displaystyle g = \left(- 7 x^{3} - 5 x^{2} + 7 x - 6\right) \left(- 6 x^{3} - 4 x^{2} + 7 x - 4\right) " src="/equation_images/%20%5Cdisplaystyle%20g%20%3D%20%5Cleft%28-%207%20x%5E%7B3%7D%20-%205%20x%5E%7B2%7D%20%2B%207%20x%20-%206%5Cright%29%20%5Cleft%28-%206%20x%5E%7B3%7D%20-%204%20x%5E%7B2%7D%20%2B%207%20x%20-%204%5Cright%29%20" alt="LaTeX: \displaystyle g = \left(- 7 x^{3} - 5 x^{2} + 7 x - 6\right) \left(- 6 x^{3} - 4 x^{2} + 7 x - 4\right) " data-equation-content=" \displaystyle g = \left(- 7 x^{3} - 5 x^{2} + 7 x - 6\right) \left(- 6 x^{3} - 4 x^{2} + 7 x - 4\right) " /> which also requires the product rule. Pushing down in the new product rule <img class="equation_image" title=" \displaystyle f=- 6 x^{3} - 4 x^{2} + 7 x - 4 \implies f'=- 18 x^{2} - 8 x + 7 " src="/equation_images/%20%5Cdisplaystyle%20f%3D-%206%20x%5E%7B3%7D%20-%204%20x%5E%7B2%7D%20%2B%207%20x%20-%204%20%5Cimplies%20f%27%3D-%2018%20x%5E%7B2%7D%20-%208%20x%20%2B%207%20" alt="LaTeX: \displaystyle f=- 6 x^{3} - 4 x^{2} + 7 x - 4 \implies f'=- 18 x^{2} - 8 x + 7 " data-equation-content=" \displaystyle f=- 6 x^{3} - 4 x^{2} + 7 x - 4 \implies f'=- 18 x^{2} - 8 x + 7 " /> and <img class="equation_image" title=" \displaystyle g=- 7 x^{3} - 5 x^{2} + 7 x - 6 \implies g'=- 21 x^{2} - 10 x + 7 " src="/equation_images/%20%5Cdisplaystyle%20g%3D-%207%20x%5E%7B3%7D%20-%205%20x%5E%7B2%7D%20%2B%207%20x%20-%206%20%5Cimplies%20g%27%3D-%2021%20x%5E%7B2%7D%20-%2010%20x%20%2B%207%20" alt="LaTeX: \displaystyle g=- 7 x^{3} - 5 x^{2} + 7 x - 6 \implies g'=- 21 x^{2} - 10 x + 7 " data-equation-content=" \displaystyle g=- 7 x^{3} - 5 x^{2} + 7 x - 6 \implies g'=- 21 x^{2} - 10 x + 7 " /> . Popping up a level gives <img class="equation_image" title=" \displaystyle g'=(- 7 x^{3} - 5 x^{2} + 7 x - 6)(- 18 x^{2} - 8 x + 7)+(- 6 x^{3} - 4 x^{2} + 7 x - 4)(- 21 x^{2} - 10 x + 7) " src="/equation_images/%20%5Cdisplaystyle%20g%27%3D%28-%207%20x%5E%7B3%7D%20-%205%20x%5E%7B2%7D%20%2B%207%20x%20-%206%29%28-%2018%20x%5E%7B2%7D%20-%208%20x%20%2B%207%29%2B%28-%206%20x%5E%7B3%7D%20-%204%20x%5E%7B2%7D%20%2B%207%20x%20-%204%29%28-%2021%20x%5E%7B2%7D%20-%2010%20x%20%2B%207%29%20" alt="LaTeX: \displaystyle g'=(- 7 x^{3} - 5 x^{2} + 7 x - 6)(- 18 x^{2} - 8 x + 7)+(- 6 x^{3} - 4 x^{2} + 7 x - 4)(- 21 x^{2} - 10 x + 7) " data-equation-content=" \displaystyle g'=(- 7 x^{3} - 5 x^{2} + 7 x - 6)(- 18 x^{2} - 8 x + 7)+(- 6 x^{3} - 4 x^{2} + 7 x - 4)(- 21 x^{2} - 10 x + 7) " /> Popping up again (Back to the original problem) gives <img class="equation_image" title=" \displaystyle f'=(- 2 x^{3} + 7 x^{2} + 6 x - 5)(\left(- 21 x^{2} - 10 x + 7\right) \left(- 6 x^{3} - 4 x^{2} + 7 x - 4\right) + \left(- 18 x^{2} - 8 x + 7\right) \left(- 7 x^{3} - 5 x^{2} + 7 x - 6\right))+(\left(- 7 x^{3} - 5 x^{2} + 7 x - 6\right) \left(- 6 x^{3} - 4 x^{2} + 7 x - 4\right))(- 6 x^{2} + 14 x + 6)=\left(- 21 x^{2} - 10 x + 7\right) \left(- 6 x^{3} - 4 x^{2} + 7 x - 4\right) \left(- 2 x^{3} + 7 x^{2} + 6 x - 5\right) + \left(- 18 x^{2} - 8 x + 7\right) \left(- 7 x^{3} - 5 x^{2} + 7 x - 6\right) \left(- 2 x^{3} + 7 x^{2} + 6 x - 5\right) + \left(- 6 x^{2} + 14 x + 6\right) \left(- 7 x^{3} - 5 x^{2} + 7 x - 6\right) \left(- 6 x^{3} - 4 x^{2} + 7 x - 4\right) " src="/equation_images/%20%5Cdisplaystyle%20f%27%3D%28-%202%20x%5E%7B3%7D%20%2B%207%20x%5E%7B2%7D%20%2B%206%20x%20-%205%29%28%5Cleft%28-%2021%20x%5E%7B2%7D%20-%2010%20x%20%2B%207%5Cright%29%20%5Cleft%28-%206%20x%5E%7B3%7D%20-%204%20x%5E%7B2%7D%20%2B%207%20x%20-%204%5Cright%29%20%2B%20%5Cleft%28-%2018%20x%5E%7B2%7D%20-%208%20x%20%2B%207%5Cright%29%20%5Cleft%28-%207%20x%5E%7B3%7D%20-%205%20x%5E%7B2%7D%20%2B%207%20x%20-%206%5Cright%29%29%2B%28%5Cleft%28-%207%20x%5E%7B3%7D%20-%205%20x%5E%7B2%7D%20%2B%207%20x%20-%206%5Cright%29%20%5Cleft%28-%206%20x%5E%7B3%7D%20-%204%20x%5E%7B2%7D%20%2B%207%20x%20-%204%5Cright%29%29%28-%206%20x%5E%7B2%7D%20%2B%2014%20x%20%2B%206%29%3D%5Cleft%28-%2021%20x%5E%7B2%7D%20-%2010%20x%20%2B%207%5Cright%29%20%5Cleft%28-%206%20x%5E%7B3%7D%20-%204%20x%5E%7B2%7D%20%2B%207%20x%20-%204%5Cright%29%20%5Cleft%28-%202%20x%5E%7B3%7D%20%2B%207%20x%5E%7B2%7D%20%2B%206%20x%20-%205%5Cright%29%20%2B%20%5Cleft%28-%2018%20x%5E%7B2%7D%20-%208%20x%20%2B%207%5Cright%29%20%5Cleft%28-%207%20x%5E%7B3%7D%20-%205%20x%5E%7B2%7D%20%2B%207%20x%20-%206%5Cright%29%20%5Cleft%28-%202%20x%5E%7B3%7D%20%2B%207%20x%5E%7B2%7D%20%2B%206%20x%20-%205%5Cright%29%20%2B%20%5Cleft%28-%206%20x%5E%7B2%7D%20%2B%2014%20x%20%2B%206%5Cright%29%20%5Cleft%28-%207%20x%5E%7B3%7D%20-%205%20x%5E%7B2%7D%20%2B%207%20x%20-%206%5Cright%29%20%5Cleft%28-%206%20x%5E%7B3%7D%20-%204%20x%5E%7B2%7D%20%2B%207%20x%20-%204%5Cright%29%20" alt="LaTeX: \displaystyle f'=(- 2 x^{3} + 7 x^{2} + 6 x - 5)(\left(- 21 x^{2} - 10 x + 7\right) \left(- 6 x^{3} - 4 x^{2} + 7 x - 4\right) + \left(- 18 x^{2} - 8 x + 7\right) \left(- 7 x^{3} - 5 x^{2} + 7 x - 6\right))+(\left(- 7 x^{3} - 5 x^{2} + 7 x - 6\right) \left(- 6 x^{3} - 4 x^{2} + 7 x - 4\right))(- 6 x^{2} + 14 x + 6)=\left(- 21 x^{2} - 10 x + 7\right) \left(- 6 x^{3} - 4 x^{2} + 7 x - 4\right) \left(- 2 x^{3} + 7 x^{2} + 6 x - 5\right) + \left(- 18 x^{2} - 8 x + 7\right) \left(- 7 x^{3} - 5 x^{2} + 7 x - 6\right) \left(- 2 x^{3} + 7 x^{2} + 6 x - 5\right) + \left(- 6 x^{2} + 14 x + 6\right) \left(- 7 x^{3} - 5 x^{2} + 7 x - 6\right) \left(- 6 x^{3} - 4 x^{2} + 7 x - 4\right) " data-equation-content=" \displaystyle f'=(- 2 x^{3} + 7 x^{2} + 6 x - 5)(\left(- 21 x^{2} - 10 x + 7\right) \left(- 6 x^{3} - 4 x^{2} + 7 x - 4\right) + \left(- 18 x^{2} - 8 x + 7\right) \left(- 7 x^{3} - 5 x^{2} + 7 x - 6\right))+(\left(- 7 x^{3} - 5 x^{2} + 7 x - 6\right) \left(- 6 x^{3} - 4 x^{2} + 7 x - 4\right))(- 6 x^{2} + 14 x + 6)=\left(- 21 x^{2} - 10 x + 7\right) \left(- 6 x^{3} - 4 x^{2} + 7 x - 4\right) \left(- 2 x^{3} + 7 x^{2} + 6 x - 5\right) + \left(- 18 x^{2} - 8 x + 7\right) \left(- 7 x^{3} - 5 x^{2} + 7 x - 6\right) \left(- 2 x^{3} + 7 x^{2} + 6 x - 5\right) + \left(- 6 x^{2} + 14 x + 6\right) \left(- 7 x^{3} - 5 x^{2} + 7 x - 6\right) \left(- 6 x^{3} - 4 x^{2} + 7 x - 4\right) " /> </p> </p>