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Find the derivative of \(\displaystyle y = (x^{3} + 5 x^{2} + 2 x - 2)(9 x^{3} + 5 x^{2} + 5 x - 4)(x^{3} - x^{2} + 5 x + 9)\).
Identifying \(\displaystyle f=x^{3} + 5 x^{2} + 2 x - 2\) and \(\displaystyle g=\left(x^{3} - x^{2} + 5 x + 9\right) \left(9 x^{3} + 5 x^{2} + 5 x - 4\right)\) and using the product rule with \(\displaystyle f=x^{3} + 5 x^{2} + 2 x - 2 \implies f'=3 x^{2} + 10 x + 2\). This leaves g as \(\displaystyle g = \left(x^{3} - x^{2} + 5 x + 9\right) \left(9 x^{3} + 5 x^{2} + 5 x - 4\right)\) which also requires the product rule. Pushing down in the new product rule \(\displaystyle f=9 x^{3} + 5 x^{2} + 5 x - 4 \implies f'=27 x^{2} + 10 x + 5\) and \(\displaystyle g=x^{3} - x^{2} + 5 x + 9 \implies g'=3 x^{2} - 2 x + 5\). Popping up a level gives \(\displaystyle g'=(x^{3} - x^{2} + 5 x + 9)(27 x^{2} + 10 x + 5)+(9 x^{3} + 5 x^{2} + 5 x - 4)(3 x^{2} - 2 x + 5)\)Popping up again (Back to the original problem) gives \(\displaystyle f'=(x^{3} + 5 x^{2} + 2 x - 2)(\left(3 x^{2} - 2 x + 5\right) \left(9 x^{3} + 5 x^{2} + 5 x - 4\right) + \left(27 x^{2} + 10 x + 5\right) \left(x^{3} - x^{2} + 5 x + 9\right))+(\left(x^{3} - x^{2} + 5 x + 9\right) \left(9 x^{3} + 5 x^{2} + 5 x - 4\right))(3 x^{2} + 10 x + 2)=\left(3 x^{2} - 2 x + 5\right) \left(x^{3} + 5 x^{2} + 2 x - 2\right) \left(9 x^{3} + 5 x^{2} + 5 x - 4\right) + \left(3 x^{2} + 10 x + 2\right) \left(x^{3} - x^{2} + 5 x + 9\right) \left(9 x^{3} + 5 x^{2} + 5 x - 4\right) + \left(27 x^{2} + 10 x + 5\right) \left(x^{3} - x^{2} + 5 x + 9\right) \left(x^{3} + 5 x^{2} + 2 x - 2\right)\)
\begin{question}Find the derivative of $y = (x^{3} + 5 x^{2} + 2 x - 2)(9 x^{3} + 5 x^{2} + 5 x - 4)(x^{3} - x^{2} + 5 x + 9)$.
\soln{9cm}{Identifying $f=x^{3} + 5 x^{2} + 2 x - 2$ and $g=\left(x^{3} - x^{2} + 5 x + 9\right) \left(9 x^{3} + 5 x^{2} + 5 x - 4\right)$ and using the product rule with $f=x^{3} + 5 x^{2} + 2 x - 2 \implies f'=3 x^{2} + 10 x + 2$. This leaves g as $g = \left(x^{3} - x^{2} + 5 x + 9\right) \left(9 x^{3} + 5 x^{2} + 5 x - 4\right)$ which also requires the product rule. Pushing down in the new product rule $f=9 x^{3} + 5 x^{2} + 5 x - 4 \implies f'=27 x^{2} + 10 x + 5$ and $g=x^{3} - x^{2} + 5 x + 9 \implies g'=3 x^{2} - 2 x + 5$. Popping up a level gives $g'=(x^{3} - x^{2} + 5 x + 9)(27 x^{2} + 10 x + 5)+(9 x^{3} + 5 x^{2} + 5 x - 4)(3 x^{2} - 2 x + 5)$Popping up again (Back to the original problem) gives $f'=(x^{3} + 5 x^{2} + 2 x - 2)(\left(3 x^{2} - 2 x + 5\right) \left(9 x^{3} + 5 x^{2} + 5 x - 4\right) + \left(27 x^{2} + 10 x + 5\right) \left(x^{3} - x^{2} + 5 x + 9\right))+(\left(x^{3} - x^{2} + 5 x + 9\right) \left(9 x^{3} + 5 x^{2} + 5 x - 4\right))(3 x^{2} + 10 x + 2)=\left(3 x^{2} - 2 x + 5\right) \left(x^{3} + 5 x^{2} + 2 x - 2\right) \left(9 x^{3} + 5 x^{2} + 5 x - 4\right) + \left(3 x^{2} + 10 x + 2\right) \left(x^{3} - x^{2} + 5 x + 9\right) \left(9 x^{3} + 5 x^{2} + 5 x - 4\right) + \left(27 x^{2} + 10 x + 5\right) \left(x^{3} - x^{2} + 5 x + 9\right) \left(x^{3} + 5 x^{2} + 2 x - 2\right)$}
\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 = (x^{3} + 5 x^{2} + 2 x - 2)(9 x^{3} + 5 x^{2} + 5 x - 4)(x^{3} - x^{2} + 5 x + 9) " src="/equation_images/%20%5Cdisplaystyle%20y%20%3D%20%28x%5E%7B3%7D%20%2B%205%20x%5E%7B2%7D%20%2B%202%20x%20-%202%29%289%20x%5E%7B3%7D%20%2B%205%20x%5E%7B2%7D%20%2B%205%20x%20-%204%29%28x%5E%7B3%7D%20-%20x%5E%7B2%7D%20%2B%205%20x%20%2B%209%29%20" alt="LaTeX: \displaystyle y = (x^{3} + 5 x^{2} + 2 x - 2)(9 x^{3} + 5 x^{2} + 5 x - 4)(x^{3} - x^{2} + 5 x + 9) " data-equation-content=" \displaystyle y = (x^{3} + 5 x^{2} + 2 x - 2)(9 x^{3} + 5 x^{2} + 5 x - 4)(x^{3} - x^{2} + 5 x + 9) " /> .</p> </p><p> <p>Identifying <img class="equation_image" title=" \displaystyle f=x^{3} + 5 x^{2} + 2 x - 2 " src="/equation_images/%20%5Cdisplaystyle%20f%3Dx%5E%7B3%7D%20%2B%205%20x%5E%7B2%7D%20%2B%202%20x%20-%202%20" alt="LaTeX: \displaystyle f=x^{3} + 5 x^{2} + 2 x - 2 " data-equation-content=" \displaystyle f=x^{3} + 5 x^{2} + 2 x - 2 " /> and <img class="equation_image" title=" \displaystyle g=\left(x^{3} - x^{2} + 5 x + 9\right) \left(9 x^{3} + 5 x^{2} + 5 x - 4\right) " src="/equation_images/%20%5Cdisplaystyle%20g%3D%5Cleft%28x%5E%7B3%7D%20-%20x%5E%7B2%7D%20%2B%205%20x%20%2B%209%5Cright%29%20%5Cleft%289%20x%5E%7B3%7D%20%2B%205%20x%5E%7B2%7D%20%2B%205%20x%20-%204%5Cright%29%20" alt="LaTeX: \displaystyle g=\left(x^{3} - x^{2} + 5 x + 9\right) \left(9 x^{3} + 5 x^{2} + 5 x - 4\right) " data-equation-content=" \displaystyle g=\left(x^{3} - x^{2} + 5 x + 9\right) \left(9 x^{3} + 5 x^{2} + 5 x - 4\right) " /> and using the product rule with <img class="equation_image" title=" \displaystyle f=x^{3} + 5 x^{2} + 2 x - 2 \implies f'=3 x^{2} + 10 x + 2 " src="/equation_images/%20%5Cdisplaystyle%20f%3Dx%5E%7B3%7D%20%2B%205%20x%5E%7B2%7D%20%2B%202%20x%20-%202%20%5Cimplies%20f%27%3D3%20x%5E%7B2%7D%20%2B%2010%20x%20%2B%202%20" alt="LaTeX: \displaystyle f=x^{3} + 5 x^{2} + 2 x - 2 \implies f'=3 x^{2} + 10 x + 2 " data-equation-content=" \displaystyle f=x^{3} + 5 x^{2} + 2 x - 2 \implies f'=3 x^{2} + 10 x + 2 " /> . This leaves g as <img class="equation_image" title=" \displaystyle g = \left(x^{3} - x^{2} + 5 x + 9\right) \left(9 x^{3} + 5 x^{2} + 5 x - 4\right) " src="/equation_images/%20%5Cdisplaystyle%20g%20%3D%20%5Cleft%28x%5E%7B3%7D%20-%20x%5E%7B2%7D%20%2B%205%20x%20%2B%209%5Cright%29%20%5Cleft%289%20x%5E%7B3%7D%20%2B%205%20x%5E%7B2%7D%20%2B%205%20x%20-%204%5Cright%29%20" alt="LaTeX: \displaystyle g = \left(x^{3} - x^{2} + 5 x + 9\right) \left(9 x^{3} + 5 x^{2} + 5 x - 4\right) " data-equation-content=" \displaystyle g = \left(x^{3} - x^{2} + 5 x + 9\right) \left(9 x^{3} + 5 x^{2} + 5 x - 4\right) " /> which also requires the product rule. Pushing down in the new product rule <img class="equation_image" title=" \displaystyle f=9 x^{3} + 5 x^{2} + 5 x - 4 \implies f'=27 x^{2} + 10 x + 5 " src="/equation_images/%20%5Cdisplaystyle%20f%3D9%20x%5E%7B3%7D%20%2B%205%20x%5E%7B2%7D%20%2B%205%20x%20-%204%20%5Cimplies%20f%27%3D27%20x%5E%7B2%7D%20%2B%2010%20x%20%2B%205%20" alt="LaTeX: \displaystyle f=9 x^{3} + 5 x^{2} + 5 x - 4 \implies f'=27 x^{2} + 10 x + 5 " data-equation-content=" \displaystyle f=9 x^{3} + 5 x^{2} + 5 x - 4 \implies f'=27 x^{2} + 10 x + 5 " /> and <img class="equation_image" title=" \displaystyle g=x^{3} - x^{2} + 5 x + 9 \implies g'=3 x^{2} - 2 x + 5 " src="/equation_images/%20%5Cdisplaystyle%20g%3Dx%5E%7B3%7D%20-%20x%5E%7B2%7D%20%2B%205%20x%20%2B%209%20%5Cimplies%20g%27%3D3%20x%5E%7B2%7D%20-%202%20x%20%2B%205%20" alt="LaTeX: \displaystyle g=x^{3} - x^{2} + 5 x + 9 \implies g'=3 x^{2} - 2 x + 5 " data-equation-content=" \displaystyle g=x^{3} - x^{2} + 5 x + 9 \implies g'=3 x^{2} - 2 x + 5 " /> . Popping up a level gives <img class="equation_image" title=" \displaystyle g'=(x^{3} - x^{2} + 5 x + 9)(27 x^{2} + 10 x + 5)+(9 x^{3} + 5 x^{2} + 5 x - 4)(3 x^{2} - 2 x + 5) " src="/equation_images/%20%5Cdisplaystyle%20g%27%3D%28x%5E%7B3%7D%20-%20x%5E%7B2%7D%20%2B%205%20x%20%2B%209%29%2827%20x%5E%7B2%7D%20%2B%2010%20x%20%2B%205%29%2B%289%20x%5E%7B3%7D%20%2B%205%20x%5E%7B2%7D%20%2B%205%20x%20-%204%29%283%20x%5E%7B2%7D%20-%202%20x%20%2B%205%29%20" alt="LaTeX: \displaystyle g'=(x^{3} - x^{2} + 5 x + 9)(27 x^{2} + 10 x + 5)+(9 x^{3} + 5 x^{2} + 5 x - 4)(3 x^{2} - 2 x + 5) " data-equation-content=" \displaystyle g'=(x^{3} - x^{2} + 5 x + 9)(27 x^{2} + 10 x + 5)+(9 x^{3} + 5 x^{2} + 5 x - 4)(3 x^{2} - 2 x + 5) " /> Popping up again (Back to the original problem) gives <img class="equation_image" title=" \displaystyle f'=(x^{3} + 5 x^{2} + 2 x - 2)(\left(3 x^{2} - 2 x + 5\right) \left(9 x^{3} + 5 x^{2} + 5 x - 4\right) + \left(27 x^{2} + 10 x + 5\right) \left(x^{3} - x^{2} + 5 x + 9\right))+(\left(x^{3} - x^{2} + 5 x + 9\right) \left(9 x^{3} + 5 x^{2} + 5 x - 4\right))(3 x^{2} + 10 x + 2)=\left(3 x^{2} - 2 x + 5\right) \left(x^{3} + 5 x^{2} + 2 x - 2\right) \left(9 x^{3} + 5 x^{2} + 5 x - 4\right) + \left(3 x^{2} + 10 x + 2\right) \left(x^{3} - x^{2} + 5 x + 9\right) \left(9 x^{3} + 5 x^{2} + 5 x - 4\right) + \left(27 x^{2} + 10 x + 5\right) \left(x^{3} - x^{2} + 5 x + 9\right) \left(x^{3} + 5 x^{2} + 2 x - 2\right) " src="/equation_images/%20%5Cdisplaystyle%20f%27%3D%28x%5E%7B3%7D%20%2B%205%20x%5E%7B2%7D%20%2B%202%20x%20-%202%29%28%5Cleft%283%20x%5E%7B2%7D%20-%202%20x%20%2B%205%5Cright%29%20%5Cleft%289%20x%5E%7B3%7D%20%2B%205%20x%5E%7B2%7D%20%2B%205%20x%20-%204%5Cright%29%20%2B%20%5Cleft%2827%20x%5E%7B2%7D%20%2B%2010%20x%20%2B%205%5Cright%29%20%5Cleft%28x%5E%7B3%7D%20-%20x%5E%7B2%7D%20%2B%205%20x%20%2B%209%5Cright%29%29%2B%28%5Cleft%28x%5E%7B3%7D%20-%20x%5E%7B2%7D%20%2B%205%20x%20%2B%209%5Cright%29%20%5Cleft%289%20x%5E%7B3%7D%20%2B%205%20x%5E%7B2%7D%20%2B%205%20x%20-%204%5Cright%29%29%283%20x%5E%7B2%7D%20%2B%2010%20x%20%2B%202%29%3D%5Cleft%283%20x%5E%7B2%7D%20-%202%20x%20%2B%205%5Cright%29%20%5Cleft%28x%5E%7B3%7D%20%2B%205%20x%5E%7B2%7D%20%2B%202%20x%20-%202%5Cright%29%20%5Cleft%289%20x%5E%7B3%7D%20%2B%205%20x%5E%7B2%7D%20%2B%205%20x%20-%204%5Cright%29%20%2B%20%5Cleft%283%20x%5E%7B2%7D%20%2B%2010%20x%20%2B%202%5Cright%29%20%5Cleft%28x%5E%7B3%7D%20-%20x%5E%7B2%7D%20%2B%205%20x%20%2B%209%5Cright%29%20%5Cleft%289%20x%5E%7B3%7D%20%2B%205%20x%5E%7B2%7D%20%2B%205%20x%20-%204%5Cright%29%20%2B%20%5Cleft%2827%20x%5E%7B2%7D%20%2B%2010%20x%20%2B%205%5Cright%29%20%5Cleft%28x%5E%7B3%7D%20-%20x%5E%7B2%7D%20%2B%205%20x%20%2B%209%5Cright%29%20%5Cleft%28x%5E%7B3%7D%20%2B%205%20x%5E%7B2%7D%20%2B%202%20x%20-%202%5Cright%29%20" alt="LaTeX: \displaystyle f'=(x^{3} + 5 x^{2} + 2 x - 2)(\left(3 x^{2} - 2 x + 5\right) \left(9 x^{3} + 5 x^{2} + 5 x - 4\right) + \left(27 x^{2} + 10 x + 5\right) \left(x^{3} - x^{2} + 5 x + 9\right))+(\left(x^{3} - x^{2} + 5 x + 9\right) \left(9 x^{3} + 5 x^{2} + 5 x - 4\right))(3 x^{2} + 10 x + 2)=\left(3 x^{2} - 2 x + 5\right) \left(x^{3} + 5 x^{2} + 2 x - 2\right) \left(9 x^{3} + 5 x^{2} + 5 x - 4\right) + \left(3 x^{2} + 10 x + 2\right) \left(x^{3} - x^{2} + 5 x + 9\right) \left(9 x^{3} + 5 x^{2} + 5 x - 4\right) + \left(27 x^{2} + 10 x + 5\right) \left(x^{3} - x^{2} + 5 x + 9\right) \left(x^{3} + 5 x^{2} + 2 x - 2\right) " data-equation-content=" \displaystyle f'=(x^{3} + 5 x^{2} + 2 x - 2)(\left(3 x^{2} - 2 x + 5\right) \left(9 x^{3} + 5 x^{2} + 5 x - 4\right) + \left(27 x^{2} + 10 x + 5\right) \left(x^{3} - x^{2} + 5 x + 9\right))+(\left(x^{3} - x^{2} + 5 x + 9\right) \left(9 x^{3} + 5 x^{2} + 5 x - 4\right))(3 x^{2} + 10 x + 2)=\left(3 x^{2} - 2 x + 5\right) \left(x^{3} + 5 x^{2} + 2 x - 2\right) \left(9 x^{3} + 5 x^{2} + 5 x - 4\right) + \left(3 x^{2} + 10 x + 2\right) \left(x^{3} - x^{2} + 5 x + 9\right) \left(9 x^{3} + 5 x^{2} + 5 x - 4\right) + \left(27 x^{2} + 10 x + 5\right) \left(x^{3} - x^{2} + 5 x + 9\right) \left(x^{3} + 5 x^{2} + 2 x - 2\right) " /> </p> </p>