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Find the derivative of \(\displaystyle y = (e^{x})(3 x^{3} + 5 x^{2} + 9 x - 7)(- 3 x^{3} + 5 x^{2} + 9 x - 2)\).
Identifying \(\displaystyle f=e^{x}\) and \(\displaystyle g=\left(- 3 x^{3} + 5 x^{2} + 9 x - 2\right) \left(3 x^{3} + 5 x^{2} + 9 x - 7\right)\) and using the product rule with \(\displaystyle f=e^{x} \implies f'=e^{x}\). This leaves g as \(\displaystyle g = \left(- 3 x^{3} + 5 x^{2} + 9 x - 2\right) \left(3 x^{3} + 5 x^{2} + 9 x - 7\right)\) which also requires the product rule. Pushing down in the new product rule \(\displaystyle f=3 x^{3} + 5 x^{2} + 9 x - 7 \implies f'=9 x^{2} + 10 x + 9\) and \(\displaystyle g=- 3 x^{3} + 5 x^{2} + 9 x - 2 \implies g'=- 9 x^{2} + 10 x + 9\). Popping up a level gives \(\displaystyle g'=(- 3 x^{3} + 5 x^{2} + 9 x - 2)(9 x^{2} + 10 x + 9)+(3 x^{3} + 5 x^{2} + 9 x - 7)(- 9 x^{2} + 10 x + 9)\)Popping up again (Back to the original problem) gives \(\displaystyle f'=(e^{x})(\left(- 9 x^{2} + 10 x + 9\right) \left(3 x^{3} + 5 x^{2} + 9 x - 7\right) + \left(9 x^{2} + 10 x + 9\right) \left(- 3 x^{3} + 5 x^{2} + 9 x - 2\right))+(\left(- 3 x^{3} + 5 x^{2} + 9 x - 2\right) \left(3 x^{3} + 5 x^{2} + 9 x - 7\right))(e^{x})=\left(- 9 x^{2} + 10 x + 9\right) \left(3 x^{3} + 5 x^{2} + 9 x - 7\right) e^{x} + \left(9 x^{2} + 10 x + 9\right) \left(- 3 x^{3} + 5 x^{2} + 9 x - 2\right) e^{x} + \left(- 3 x^{3} + 5 x^{2} + 9 x - 2\right) \left(3 x^{3} + 5 x^{2} + 9 x - 7\right) e^{x}\)
\begin{question}Find the derivative of $y = (e^{x})(3 x^{3} + 5 x^{2} + 9 x - 7)(- 3 x^{3} + 5 x^{2} + 9 x - 2)$.
\soln{9cm}{Identifying $f=e^{x}$ and $g=\left(- 3 x^{3} + 5 x^{2} + 9 x - 2\right) \left(3 x^{3} + 5 x^{2} + 9 x - 7\right)$ and using the product rule with $f=e^{x} \implies f'=e^{x}$. This leaves g as $g = \left(- 3 x^{3} + 5 x^{2} + 9 x - 2\right) \left(3 x^{3} + 5 x^{2} + 9 x - 7\right)$ which also requires the product rule. Pushing down in the new product rule $f=3 x^{3} + 5 x^{2} + 9 x - 7 \implies f'=9 x^{2} + 10 x + 9$ and $g=- 3 x^{3} + 5 x^{2} + 9 x - 2 \implies g'=- 9 x^{2} + 10 x + 9$. Popping up a level gives $g'=(- 3 x^{3} + 5 x^{2} + 9 x - 2)(9 x^{2} + 10 x + 9)+(3 x^{3} + 5 x^{2} + 9 x - 7)(- 9 x^{2} + 10 x + 9)$Popping up again (Back to the original problem) gives $f'=(e^{x})(\left(- 9 x^{2} + 10 x + 9\right) \left(3 x^{3} + 5 x^{2} + 9 x - 7\right) + \left(9 x^{2} + 10 x + 9\right) \left(- 3 x^{3} + 5 x^{2} + 9 x - 2\right))+(\left(- 3 x^{3} + 5 x^{2} + 9 x - 2\right) \left(3 x^{3} + 5 x^{2} + 9 x - 7\right))(e^{x})=\left(- 9 x^{2} + 10 x + 9\right) \left(3 x^{3} + 5 x^{2} + 9 x - 7\right) e^{x} + \left(9 x^{2} + 10 x + 9\right) \left(- 3 x^{3} + 5 x^{2} + 9 x - 2\right) e^{x} + \left(- 3 x^{3} + 5 x^{2} + 9 x - 2\right) \left(3 x^{3} + 5 x^{2} + 9 x - 7\right) e^{x}$}
\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 = (e^{x})(3 x^{3} + 5 x^{2} + 9 x - 7)(- 3 x^{3} + 5 x^{2} + 9 x - 2) " src="/equation_images/%20%5Cdisplaystyle%20y%20%3D%20%28e%5E%7Bx%7D%29%283%20x%5E%7B3%7D%20%2B%205%20x%5E%7B2%7D%20%2B%209%20x%20-%207%29%28-%203%20x%5E%7B3%7D%20%2B%205%20x%5E%7B2%7D%20%2B%209%20x%20-%202%29%20" alt="LaTeX: \displaystyle y = (e^{x})(3 x^{3} + 5 x^{2} + 9 x - 7)(- 3 x^{3} + 5 x^{2} + 9 x - 2) " data-equation-content=" \displaystyle y = (e^{x})(3 x^{3} + 5 x^{2} + 9 x - 7)(- 3 x^{3} + 5 x^{2} + 9 x - 2) " /> .</p> </p><p> <p>Identifying <img class="equation_image" title=" \displaystyle f=e^{x} " src="/equation_images/%20%5Cdisplaystyle%20f%3De%5E%7Bx%7D%20" alt="LaTeX: \displaystyle f=e^{x} " data-equation-content=" \displaystyle f=e^{x} " /> and <img class="equation_image" title=" \displaystyle g=\left(- 3 x^{3} + 5 x^{2} + 9 x - 2\right) \left(3 x^{3} + 5 x^{2} + 9 x - 7\right) " src="/equation_images/%20%5Cdisplaystyle%20g%3D%5Cleft%28-%203%20x%5E%7B3%7D%20%2B%205%20x%5E%7B2%7D%20%2B%209%20x%20-%202%5Cright%29%20%5Cleft%283%20x%5E%7B3%7D%20%2B%205%20x%5E%7B2%7D%20%2B%209%20x%20-%207%5Cright%29%20" alt="LaTeX: \displaystyle g=\left(- 3 x^{3} + 5 x^{2} + 9 x - 2\right) \left(3 x^{3} + 5 x^{2} + 9 x - 7\right) " data-equation-content=" \displaystyle g=\left(- 3 x^{3} + 5 x^{2} + 9 x - 2\right) \left(3 x^{3} + 5 x^{2} + 9 x - 7\right) " /> and using the product rule with <img class="equation_image" title=" \displaystyle f=e^{x} \implies f'=e^{x} " src="/equation_images/%20%5Cdisplaystyle%20f%3De%5E%7Bx%7D%20%5Cimplies%20f%27%3De%5E%7Bx%7D%20" alt="LaTeX: \displaystyle f=e^{x} \implies f'=e^{x} " data-equation-content=" \displaystyle f=e^{x} \implies f'=e^{x} " /> . This leaves g as <img class="equation_image" title=" \displaystyle g = \left(- 3 x^{3} + 5 x^{2} + 9 x - 2\right) \left(3 x^{3} + 5 x^{2} + 9 x - 7\right) " src="/equation_images/%20%5Cdisplaystyle%20g%20%3D%20%5Cleft%28-%203%20x%5E%7B3%7D%20%2B%205%20x%5E%7B2%7D%20%2B%209%20x%20-%202%5Cright%29%20%5Cleft%283%20x%5E%7B3%7D%20%2B%205%20x%5E%7B2%7D%20%2B%209%20x%20-%207%5Cright%29%20" alt="LaTeX: \displaystyle g = \left(- 3 x^{3} + 5 x^{2} + 9 x - 2\right) \left(3 x^{3} + 5 x^{2} + 9 x - 7\right) " data-equation-content=" \displaystyle g = \left(- 3 x^{3} + 5 x^{2} + 9 x - 2\right) \left(3 x^{3} + 5 x^{2} + 9 x - 7\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 - 7 \implies f'=9 x^{2} + 10 x + 9 " src="/equation_images/%20%5Cdisplaystyle%20f%3D3%20x%5E%7B3%7D%20%2B%205%20x%5E%7B2%7D%20%2B%209%20x%20-%207%20%5Cimplies%20f%27%3D9%20x%5E%7B2%7D%20%2B%2010%20x%20%2B%209%20" alt="LaTeX: \displaystyle f=3 x^{3} + 5 x^{2} + 9 x - 7 \implies f'=9 x^{2} + 10 x + 9 " data-equation-content=" \displaystyle f=3 x^{3} + 5 x^{2} + 9 x - 7 \implies f'=9 x^{2} + 10 x + 9 " /> and <img class="equation_image" title=" \displaystyle g=- 3 x^{3} + 5 x^{2} + 9 x - 2 \implies g'=- 9 x^{2} + 10 x + 9 " src="/equation_images/%20%5Cdisplaystyle%20g%3D-%203%20x%5E%7B3%7D%20%2B%205%20x%5E%7B2%7D%20%2B%209%20x%20-%202%20%5Cimplies%20g%27%3D-%209%20x%5E%7B2%7D%20%2B%2010%20x%20%2B%209%20" alt="LaTeX: \displaystyle g=- 3 x^{3} + 5 x^{2} + 9 x - 2 \implies g'=- 9 x^{2} + 10 x + 9 " data-equation-content=" \displaystyle g=- 3 x^{3} + 5 x^{2} + 9 x - 2 \implies g'=- 9 x^{2} + 10 x + 9 " /> . Popping up a level gives <img class="equation_image" title=" \displaystyle g'=(- 3 x^{3} + 5 x^{2} + 9 x - 2)(9 x^{2} + 10 x + 9)+(3 x^{3} + 5 x^{2} + 9 x - 7)(- 9 x^{2} + 10 x + 9) " src="/equation_images/%20%5Cdisplaystyle%20g%27%3D%28-%203%20x%5E%7B3%7D%20%2B%205%20x%5E%7B2%7D%20%2B%209%20x%20-%202%29%289%20x%5E%7B2%7D%20%2B%2010%20x%20%2B%209%29%2B%283%20x%5E%7B3%7D%20%2B%205%20x%5E%7B2%7D%20%2B%209%20x%20-%207%29%28-%209%20x%5E%7B2%7D%20%2B%2010%20x%20%2B%209%29%20" alt="LaTeX: \displaystyle g'=(- 3 x^{3} + 5 x^{2} + 9 x - 2)(9 x^{2} + 10 x + 9)+(3 x^{3} + 5 x^{2} + 9 x - 7)(- 9 x^{2} + 10 x + 9) " data-equation-content=" \displaystyle g'=(- 3 x^{3} + 5 x^{2} + 9 x - 2)(9 x^{2} + 10 x + 9)+(3 x^{3} + 5 x^{2} + 9 x - 7)(- 9 x^{2} + 10 x + 9) " /> Popping up again (Back to the original problem) gives <img class="equation_image" title=" \displaystyle f'=(e^{x})(\left(- 9 x^{2} + 10 x + 9\right) \left(3 x^{3} + 5 x^{2} + 9 x - 7\right) + \left(9 x^{2} + 10 x + 9\right) \left(- 3 x^{3} + 5 x^{2} + 9 x - 2\right))+(\left(- 3 x^{3} + 5 x^{2} + 9 x - 2\right) \left(3 x^{3} + 5 x^{2} + 9 x - 7\right))(e^{x})=\left(- 9 x^{2} + 10 x + 9\right) \left(3 x^{3} + 5 x^{2} + 9 x - 7\right) e^{x} + \left(9 x^{2} + 10 x + 9\right) \left(- 3 x^{3} + 5 x^{2} + 9 x - 2\right) e^{x} + \left(- 3 x^{3} + 5 x^{2} + 9 x - 2\right) \left(3 x^{3} + 5 x^{2} + 9 x - 7\right) e^{x} " src="/equation_images/%20%5Cdisplaystyle%20f%27%3D%28e%5E%7Bx%7D%29%28%5Cleft%28-%209%20x%5E%7B2%7D%20%2B%2010%20x%20%2B%209%5Cright%29%20%5Cleft%283%20x%5E%7B3%7D%20%2B%205%20x%5E%7B2%7D%20%2B%209%20x%20-%207%5Cright%29%20%2B%20%5Cleft%289%20x%5E%7B2%7D%20%2B%2010%20x%20%2B%209%5Cright%29%20%5Cleft%28-%203%20x%5E%7B3%7D%20%2B%205%20x%5E%7B2%7D%20%2B%209%20x%20-%202%5Cright%29%29%2B%28%5Cleft%28-%203%20x%5E%7B3%7D%20%2B%205%20x%5E%7B2%7D%20%2B%209%20x%20-%202%5Cright%29%20%5Cleft%283%20x%5E%7B3%7D%20%2B%205%20x%5E%7B2%7D%20%2B%209%20x%20-%207%5Cright%29%29%28e%5E%7Bx%7D%29%3D%5Cleft%28-%209%20x%5E%7B2%7D%20%2B%2010%20x%20%2B%209%5Cright%29%20%5Cleft%283%20x%5E%7B3%7D%20%2B%205%20x%5E%7B2%7D%20%2B%209%20x%20-%207%5Cright%29%20e%5E%7Bx%7D%20%2B%20%5Cleft%289%20x%5E%7B2%7D%20%2B%2010%20x%20%2B%209%5Cright%29%20%5Cleft%28-%203%20x%5E%7B3%7D%20%2B%205%20x%5E%7B2%7D%20%2B%209%20x%20-%202%5Cright%29%20e%5E%7Bx%7D%20%2B%20%5Cleft%28-%203%20x%5E%7B3%7D%20%2B%205%20x%5E%7B2%7D%20%2B%209%20x%20-%202%5Cright%29%20%5Cleft%283%20x%5E%7B3%7D%20%2B%205%20x%5E%7B2%7D%20%2B%209%20x%20-%207%5Cright%29%20e%5E%7Bx%7D%20" alt="LaTeX: \displaystyle f'=(e^{x})(\left(- 9 x^{2} + 10 x + 9\right) \left(3 x^{3} + 5 x^{2} + 9 x - 7\right) + \left(9 x^{2} + 10 x + 9\right) \left(- 3 x^{3} + 5 x^{2} + 9 x - 2\right))+(\left(- 3 x^{3} + 5 x^{2} + 9 x - 2\right) \left(3 x^{3} + 5 x^{2} + 9 x - 7\right))(e^{x})=\left(- 9 x^{2} + 10 x + 9\right) \left(3 x^{3} + 5 x^{2} + 9 x - 7\right) e^{x} + \left(9 x^{2} + 10 x + 9\right) \left(- 3 x^{3} + 5 x^{2} + 9 x - 2\right) e^{x} + \left(- 3 x^{3} + 5 x^{2} + 9 x - 2\right) \left(3 x^{3} + 5 x^{2} + 9 x - 7\right) e^{x} " data-equation-content=" \displaystyle f'=(e^{x})(\left(- 9 x^{2} + 10 x + 9\right) \left(3 x^{3} + 5 x^{2} + 9 x - 7\right) + \left(9 x^{2} + 10 x + 9\right) \left(- 3 x^{3} + 5 x^{2} + 9 x - 2\right))+(\left(- 3 x^{3} + 5 x^{2} + 9 x - 2\right) \left(3 x^{3} + 5 x^{2} + 9 x - 7\right))(e^{x})=\left(- 9 x^{2} + 10 x + 9\right) \left(3 x^{3} + 5 x^{2} + 9 x - 7\right) e^{x} + \left(9 x^{2} + 10 x + 9\right) \left(- 3 x^{3} + 5 x^{2} + 9 x - 2\right) e^{x} + \left(- 3 x^{3} + 5 x^{2} + 9 x - 2\right) \left(3 x^{3} + 5 x^{2} + 9 x - 7\right) e^{x} " /> </p> </p>