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Find the derivative of \(\displaystyle y = (7 x^{3} - 7 x^{2} + 5 x - 7)(\sin{\left(x \right)})(- x^{3} - 2 x^{2} - 9 x + 6)\).
Identifying \(\displaystyle f=7 x^{3} - 7 x^{2} + 5 x - 7\) and \(\displaystyle g=\left(- x^{3} - 2 x^{2} - 9 x + 6\right) \sin{\left(x \right)}\) and using the product rule with \(\displaystyle f=7 x^{3} - 7 x^{2} + 5 x - 7 \implies f'=21 x^{2} - 14 x + 5\). This leaves g as \(\displaystyle g = \left(- x^{3} - 2 x^{2} - 9 x + 6\right) \sin{\left(x \right)}\) which also requires the product rule. Pushing down in the new product rule \(\displaystyle f=\sin{\left(x \right)} \implies f'=\cos{\left(x \right)}\) and \(\displaystyle g=- x^{3} - 2 x^{2} - 9 x + 6 \implies g'=- 3 x^{2} - 4 x - 9\). Popping up a level gives \(\displaystyle g'=(- x^{3} - 2 x^{2} - 9 x + 6)(\cos{\left(x \right)})+(\sin{\left(x \right)})(- 3 x^{2} - 4 x - 9)\)Popping up again (Back to the original problem) gives \(\displaystyle f'=(7 x^{3} - 7 x^{2} + 5 x - 7)(\left(- 3 x^{2} - 4 x - 9\right) \sin{\left(x \right)} + \left(- x^{3} - 2 x^{2} - 9 x + 6\right) \cos{\left(x \right)})+(\left(- x^{3} - 2 x^{2} - 9 x + 6\right) \sin{\left(x \right)})(21 x^{2} - 14 x + 5)=\left(- 3 x^{2} - 4 x - 9\right) \left(7 x^{3} - 7 x^{2} + 5 x - 7\right) \sin{\left(x \right)} + \left(21 x^{2} - 14 x + 5\right) \left(- x^{3} - 2 x^{2} - 9 x + 6\right) \sin{\left(x \right)} + \left(- x^{3} - 2 x^{2} - 9 x + 6\right) \left(7 x^{3} - 7 x^{2} + 5 x - 7\right) \cos{\left(x \right)}\)
\begin{question}Find the derivative of $y = (7 x^{3} - 7 x^{2} + 5 x - 7)(\sin{\left(x \right)})(- x^{3} - 2 x^{2} - 9 x + 6)$.
\soln{9cm}{Identifying $f=7 x^{3} - 7 x^{2} + 5 x - 7$ and $g=\left(- x^{3} - 2 x^{2} - 9 x + 6\right) \sin{\left(x \right)}$ and using the product rule with $f=7 x^{3} - 7 x^{2} + 5 x - 7 \implies f'=21 x^{2} - 14 x + 5$. This leaves g as $g = \left(- x^{3} - 2 x^{2} - 9 x + 6\right) \sin{\left(x \right)}$ which also requires the product rule. Pushing down in the new product rule $f=\sin{\left(x \right)} \implies f'=\cos{\left(x \right)}$ and $g=- x^{3} - 2 x^{2} - 9 x + 6 \implies g'=- 3 x^{2} - 4 x - 9$. Popping up a level gives $g'=(- x^{3} - 2 x^{2} - 9 x + 6)(\cos{\left(x \right)})+(\sin{\left(x \right)})(- 3 x^{2} - 4 x - 9)$Popping up again (Back to the original problem) gives $f'=(7 x^{3} - 7 x^{2} + 5 x - 7)(\left(- 3 x^{2} - 4 x - 9\right) \sin{\left(x \right)} + \left(- x^{3} - 2 x^{2} - 9 x + 6\right) \cos{\left(x \right)})+(\left(- x^{3} - 2 x^{2} - 9 x + 6\right) \sin{\left(x \right)})(21 x^{2} - 14 x + 5)=\left(- 3 x^{2} - 4 x - 9\right) \left(7 x^{3} - 7 x^{2} + 5 x - 7\right) \sin{\left(x \right)} + \left(21 x^{2} - 14 x + 5\right) \left(- x^{3} - 2 x^{2} - 9 x + 6\right) \sin{\left(x \right)} + \left(- x^{3} - 2 x^{2} - 9 x + 6\right) \left(7 x^{3} - 7 x^{2} + 5 x - 7\right) \cos{\left(x \right)}$}
\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 = (7 x^{3} - 7 x^{2} + 5 x - 7)(\sin{\left(x \right)})(- x^{3} - 2 x^{2} - 9 x + 6) " src="/equation_images/%20%5Cdisplaystyle%20y%20%3D%20%287%20x%5E%7B3%7D%20-%207%20x%5E%7B2%7D%20%2B%205%20x%20-%207%29%28%5Csin%7B%5Cleft%28x%20%5Cright%29%7D%29%28-%20x%5E%7B3%7D%20-%202%20x%5E%7B2%7D%20-%209%20x%20%2B%206%29%20" alt="LaTeX: \displaystyle y = (7 x^{3} - 7 x^{2} + 5 x - 7)(\sin{\left(x \right)})(- x^{3} - 2 x^{2} - 9 x + 6) " data-equation-content=" \displaystyle y = (7 x^{3} - 7 x^{2} + 5 x - 7)(\sin{\left(x \right)})(- x^{3} - 2 x^{2} - 9 x + 6) " /> .</p> </p><p> <p>Identifying <img class="equation_image" title=" \displaystyle f=7 x^{3} - 7 x^{2} + 5 x - 7 " src="/equation_images/%20%5Cdisplaystyle%20f%3D7%20x%5E%7B3%7D%20-%207%20x%5E%7B2%7D%20%2B%205%20x%20-%207%20" alt="LaTeX: \displaystyle f=7 x^{3} - 7 x^{2} + 5 x - 7 " data-equation-content=" \displaystyle f=7 x^{3} - 7 x^{2} + 5 x - 7 " /> and <img class="equation_image" title=" \displaystyle g=\left(- x^{3} - 2 x^{2} - 9 x + 6\right) \sin{\left(x \right)} " src="/equation_images/%20%5Cdisplaystyle%20g%3D%5Cleft%28-%20x%5E%7B3%7D%20-%202%20x%5E%7B2%7D%20-%209%20x%20%2B%206%5Cright%29%20%5Csin%7B%5Cleft%28x%20%5Cright%29%7D%20" alt="LaTeX: \displaystyle g=\left(- x^{3} - 2 x^{2} - 9 x + 6\right) \sin{\left(x \right)} " data-equation-content=" \displaystyle g=\left(- x^{3} - 2 x^{2} - 9 x + 6\right) \sin{\left(x \right)} " /> and using the product rule with <img class="equation_image" title=" \displaystyle f=7 x^{3} - 7 x^{2} + 5 x - 7 \implies f'=21 x^{2} - 14 x + 5 " src="/equation_images/%20%5Cdisplaystyle%20f%3D7%20x%5E%7B3%7D%20-%207%20x%5E%7B2%7D%20%2B%205%20x%20-%207%20%5Cimplies%20f%27%3D21%20x%5E%7B2%7D%20-%2014%20x%20%2B%205%20" alt="LaTeX: \displaystyle f=7 x^{3} - 7 x^{2} + 5 x - 7 \implies f'=21 x^{2} - 14 x + 5 " data-equation-content=" \displaystyle f=7 x^{3} - 7 x^{2} + 5 x - 7 \implies f'=21 x^{2} - 14 x + 5 " /> . This leaves g as <img class="equation_image" title=" \displaystyle g = \left(- x^{3} - 2 x^{2} - 9 x + 6\right) \sin{\left(x \right)} " src="/equation_images/%20%5Cdisplaystyle%20g%20%3D%20%5Cleft%28-%20x%5E%7B3%7D%20-%202%20x%5E%7B2%7D%20-%209%20x%20%2B%206%5Cright%29%20%5Csin%7B%5Cleft%28x%20%5Cright%29%7D%20" alt="LaTeX: \displaystyle g = \left(- x^{3} - 2 x^{2} - 9 x + 6\right) \sin{\left(x \right)} " data-equation-content=" \displaystyle g = \left(- x^{3} - 2 x^{2} - 9 x + 6\right) \sin{\left(x \right)} " /> which also requires the product rule. Pushing down in the new product rule <img class="equation_image" title=" \displaystyle f=\sin{\left(x \right)} \implies f'=\cos{\left(x \right)} " src="/equation_images/%20%5Cdisplaystyle%20f%3D%5Csin%7B%5Cleft%28x%20%5Cright%29%7D%20%5Cimplies%20f%27%3D%5Ccos%7B%5Cleft%28x%20%5Cright%29%7D%20" alt="LaTeX: \displaystyle f=\sin{\left(x \right)} \implies f'=\cos{\left(x \right)} " data-equation-content=" \displaystyle f=\sin{\left(x \right)} \implies f'=\cos{\left(x \right)} " /> and <img class="equation_image" title=" \displaystyle g=- x^{3} - 2 x^{2} - 9 x + 6 \implies g'=- 3 x^{2} - 4 x - 9 " src="/equation_images/%20%5Cdisplaystyle%20g%3D-%20x%5E%7B3%7D%20-%202%20x%5E%7B2%7D%20-%209%20x%20%2B%206%20%5Cimplies%20g%27%3D-%203%20x%5E%7B2%7D%20-%204%20x%20-%209%20" alt="LaTeX: \displaystyle g=- x^{3} - 2 x^{2} - 9 x + 6 \implies g'=- 3 x^{2} - 4 x - 9 " data-equation-content=" \displaystyle g=- x^{3} - 2 x^{2} - 9 x + 6 \implies g'=- 3 x^{2} - 4 x - 9 " /> . Popping up a level gives <img class="equation_image" title=" \displaystyle g'=(- x^{3} - 2 x^{2} - 9 x + 6)(\cos{\left(x \right)})+(\sin{\left(x \right)})(- 3 x^{2} - 4 x - 9) " src="/equation_images/%20%5Cdisplaystyle%20g%27%3D%28-%20x%5E%7B3%7D%20-%202%20x%5E%7B2%7D%20-%209%20x%20%2B%206%29%28%5Ccos%7B%5Cleft%28x%20%5Cright%29%7D%29%2B%28%5Csin%7B%5Cleft%28x%20%5Cright%29%7D%29%28-%203%20x%5E%7B2%7D%20-%204%20x%20-%209%29%20" alt="LaTeX: \displaystyle g'=(- x^{3} - 2 x^{2} - 9 x + 6)(\cos{\left(x \right)})+(\sin{\left(x \right)})(- 3 x^{2} - 4 x - 9) " data-equation-content=" \displaystyle g'=(- x^{3} - 2 x^{2} - 9 x + 6)(\cos{\left(x \right)})+(\sin{\left(x \right)})(- 3 x^{2} - 4 x - 9) " /> Popping up again (Back to the original problem) gives <img class="equation_image" title=" \displaystyle f'=(7 x^{3} - 7 x^{2} + 5 x - 7)(\left(- 3 x^{2} - 4 x - 9\right) \sin{\left(x \right)} + \left(- x^{3} - 2 x^{2} - 9 x + 6\right) \cos{\left(x \right)})+(\left(- x^{3} - 2 x^{2} - 9 x + 6\right) \sin{\left(x \right)})(21 x^{2} - 14 x + 5)=\left(- 3 x^{2} - 4 x - 9\right) \left(7 x^{3} - 7 x^{2} + 5 x - 7\right) \sin{\left(x \right)} + \left(21 x^{2} - 14 x + 5\right) \left(- x^{3} - 2 x^{2} - 9 x + 6\right) \sin{\left(x \right)} + \left(- x^{3} - 2 x^{2} - 9 x + 6\right) \left(7 x^{3} - 7 x^{2} + 5 x - 7\right) \cos{\left(x \right)} " src="/equation_images/%20%5Cdisplaystyle%20f%27%3D%287%20x%5E%7B3%7D%20-%207%20x%5E%7B2%7D%20%2B%205%20x%20-%207%29%28%5Cleft%28-%203%20x%5E%7B2%7D%20-%204%20x%20-%209%5Cright%29%20%5Csin%7B%5Cleft%28x%20%5Cright%29%7D%20%2B%20%5Cleft%28-%20x%5E%7B3%7D%20-%202%20x%5E%7B2%7D%20-%209%20x%20%2B%206%5Cright%29%20%5Ccos%7B%5Cleft%28x%20%5Cright%29%7D%29%2B%28%5Cleft%28-%20x%5E%7B3%7D%20-%202%20x%5E%7B2%7D%20-%209%20x%20%2B%206%5Cright%29%20%5Csin%7B%5Cleft%28x%20%5Cright%29%7D%29%2821%20x%5E%7B2%7D%20-%2014%20x%20%2B%205%29%3D%5Cleft%28-%203%20x%5E%7B2%7D%20-%204%20x%20-%209%5Cright%29%20%5Cleft%287%20x%5E%7B3%7D%20-%207%20x%5E%7B2%7D%20%2B%205%20x%20-%207%5Cright%29%20%5Csin%7B%5Cleft%28x%20%5Cright%29%7D%20%2B%20%5Cleft%2821%20x%5E%7B2%7D%20-%2014%20x%20%2B%205%5Cright%29%20%5Cleft%28-%20x%5E%7B3%7D%20-%202%20x%5E%7B2%7D%20-%209%20x%20%2B%206%5Cright%29%20%5Csin%7B%5Cleft%28x%20%5Cright%29%7D%20%2B%20%5Cleft%28-%20x%5E%7B3%7D%20-%202%20x%5E%7B2%7D%20-%209%20x%20%2B%206%5Cright%29%20%5Cleft%287%20x%5E%7B3%7D%20-%207%20x%5E%7B2%7D%20%2B%205%20x%20-%207%5Cright%29%20%5Ccos%7B%5Cleft%28x%20%5Cright%29%7D%20" alt="LaTeX: \displaystyle f'=(7 x^{3} - 7 x^{2} + 5 x - 7)(\left(- 3 x^{2} - 4 x - 9\right) \sin{\left(x \right)} + \left(- x^{3} - 2 x^{2} - 9 x + 6\right) \cos{\left(x \right)})+(\left(- x^{3} - 2 x^{2} - 9 x + 6\right) \sin{\left(x \right)})(21 x^{2} - 14 x + 5)=\left(- 3 x^{2} - 4 x - 9\right) \left(7 x^{3} - 7 x^{2} + 5 x - 7\right) \sin{\left(x \right)} + \left(21 x^{2} - 14 x + 5\right) \left(- x^{3} - 2 x^{2} - 9 x + 6\right) \sin{\left(x \right)} + \left(- x^{3} - 2 x^{2} - 9 x + 6\right) \left(7 x^{3} - 7 x^{2} + 5 x - 7\right) \cos{\left(x \right)} " data-equation-content=" \displaystyle f'=(7 x^{3} - 7 x^{2} + 5 x - 7)(\left(- 3 x^{2} - 4 x - 9\right) \sin{\left(x \right)} + \left(- x^{3} - 2 x^{2} - 9 x + 6\right) \cos{\left(x \right)})+(\left(- x^{3} - 2 x^{2} - 9 x + 6\right) \sin{\left(x \right)})(21 x^{2} - 14 x + 5)=\left(- 3 x^{2} - 4 x - 9\right) \left(7 x^{3} - 7 x^{2} + 5 x - 7\right) \sin{\left(x \right)} + \left(21 x^{2} - 14 x + 5\right) \left(- x^{3} - 2 x^{2} - 9 x + 6\right) \sin{\left(x \right)} + \left(- x^{3} - 2 x^{2} - 9 x + 6\right) \left(7 x^{3} - 7 x^{2} + 5 x - 7\right) \cos{\left(x \right)} " /> </p> </p>