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Questions: Algebra BusinessCalculus
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Find the derivative of \(\displaystyle y = (5 x^{2} + 7 x + 2)(5 x^{2} - 4 x + 2)(7 x^{2} + 5 x + 3)\).
Identifying \(\displaystyle f=5 x^{2} + 7 x + 2\) and \(\displaystyle g=\left(5 x^{2} - 4 x + 2\right) \left(7 x^{2} + 5 x + 3\right)\) and using the product rule with \(\displaystyle f=5 x^{2} + 7 x + 2 \implies f'=10 x + 7\). This leaves g as \(\displaystyle g = \left(5 x^{2} - 4 x + 2\right) \left(7 x^{2} + 5 x + 3\right)\) which also requires the product rule. Pushing down in the new product rule \(\displaystyle f=5 x^{2} - 4 x + 2 \implies f'=10 x - 4\) and \(\displaystyle g=7 x^{2} + 5 x + 3 \implies g'=14 x + 5\). Popping up a level gives \(\displaystyle g'=(7 x^{2} + 5 x + 3)(10 x - 4)+(5 x^{2} - 4 x + 2)(14 x + 5)\)Popping up again (Back to the original problem) gives \(\displaystyle f'=(5 x^{2} + 7 x + 2)(\left(10 x - 4\right) \left(7 x^{2} + 5 x + 3\right) + \left(14 x + 5\right) \left(5 x^{2} - 4 x + 2\right))+(\left(5 x^{2} - 4 x + 2\right) \left(7 x^{2} + 5 x + 3\right))(10 x + 7)=\left(10 x - 4\right) \left(5 x^{2} + 7 x + 2\right) \left(7 x^{2} + 5 x + 3\right) + \left(10 x + 7\right) \left(5 x^{2} - 4 x + 2\right) \left(7 x^{2} + 5 x + 3\right) + \left(14 x + 5\right) \left(5 x^{2} - 4 x + 2\right) \left(5 x^{2} + 7 x + 2\right)\)
\begin{question}Find the derivative of $y = (5 x^{2} + 7 x + 2)(5 x^{2} - 4 x + 2)(7 x^{2} + 5 x + 3)$.
\soln{9cm}{Identifying $f=5 x^{2} + 7 x + 2$ and $g=\left(5 x^{2} - 4 x + 2\right) \left(7 x^{2} + 5 x + 3\right)$ and using the product rule with $f=5 x^{2} + 7 x + 2 \implies f'=10 x + 7$. This leaves g as $g = \left(5 x^{2} - 4 x + 2\right) \left(7 x^{2} + 5 x + 3\right)$ which also requires the product rule. Pushing down in the new product rule $f=5 x^{2} - 4 x + 2 \implies f'=10 x - 4$ and $g=7 x^{2} + 5 x + 3 \implies g'=14 x + 5$. Popping up a level gives $g'=(7 x^{2} + 5 x + 3)(10 x - 4)+(5 x^{2} - 4 x + 2)(14 x + 5)$Popping up again (Back to the original problem) gives $f'=(5 x^{2} + 7 x + 2)(\left(10 x - 4\right) \left(7 x^{2} + 5 x + 3\right) + \left(14 x + 5\right) \left(5 x^{2} - 4 x + 2\right))+(\left(5 x^{2} - 4 x + 2\right) \left(7 x^{2} + 5 x + 3\right))(10 x + 7)=\left(10 x - 4\right) \left(5 x^{2} + 7 x + 2\right) \left(7 x^{2} + 5 x + 3\right) + \left(10 x + 7\right) \left(5 x^{2} - 4 x + 2\right) \left(7 x^{2} + 5 x + 3\right) + \left(14 x + 5\right) \left(5 x^{2} - 4 x + 2\right) \left(5 x^{2} + 7 x + 2\right)$}
\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 = (5 x^{2} + 7 x + 2)(5 x^{2} - 4 x + 2)(7 x^{2} + 5 x + 3) " src="/equation_images/%20%5Cdisplaystyle%20y%20%3D%20%285%20x%5E%7B2%7D%20%2B%207%20x%20%2B%202%29%285%20x%5E%7B2%7D%20-%204%20x%20%2B%202%29%287%20x%5E%7B2%7D%20%2B%205%20x%20%2B%203%29%20" alt="LaTeX: \displaystyle y = (5 x^{2} + 7 x + 2)(5 x^{2} - 4 x + 2)(7 x^{2} + 5 x + 3) " data-equation-content=" \displaystyle y = (5 x^{2} + 7 x + 2)(5 x^{2} - 4 x + 2)(7 x^{2} + 5 x + 3) " /> .</p> </p><p> <p>Identifying <img class="equation_image" title=" \displaystyle f=5 x^{2} + 7 x + 2 " src="/equation_images/%20%5Cdisplaystyle%20f%3D5%20x%5E%7B2%7D%20%2B%207%20x%20%2B%202%20" alt="LaTeX: \displaystyle f=5 x^{2} + 7 x + 2 " data-equation-content=" \displaystyle f=5 x^{2} + 7 x + 2 " /> and <img class="equation_image" title=" \displaystyle g=\left(5 x^{2} - 4 x + 2\right) \left(7 x^{2} + 5 x + 3\right) " src="/equation_images/%20%5Cdisplaystyle%20g%3D%5Cleft%285%20x%5E%7B2%7D%20-%204%20x%20%2B%202%5Cright%29%20%5Cleft%287%20x%5E%7B2%7D%20%2B%205%20x%20%2B%203%5Cright%29%20" alt="LaTeX: \displaystyle g=\left(5 x^{2} - 4 x + 2\right) \left(7 x^{2} + 5 x + 3\right) " data-equation-content=" \displaystyle g=\left(5 x^{2} - 4 x + 2\right) \left(7 x^{2} + 5 x + 3\right) " /> and using the product rule with <img class="equation_image" title=" \displaystyle f=5 x^{2} + 7 x + 2 \implies f'=10 x + 7 " src="/equation_images/%20%5Cdisplaystyle%20f%3D5%20x%5E%7B2%7D%20%2B%207%20x%20%2B%202%20%5Cimplies%20f%27%3D10%20x%20%2B%207%20" alt="LaTeX: \displaystyle f=5 x^{2} + 7 x + 2 \implies f'=10 x + 7 " data-equation-content=" \displaystyle f=5 x^{2} + 7 x + 2 \implies f'=10 x + 7 " /> . This leaves g as <img class="equation_image" title=" \displaystyle g = \left(5 x^{2} - 4 x + 2\right) \left(7 x^{2} + 5 x + 3\right) " src="/equation_images/%20%5Cdisplaystyle%20g%20%3D%20%5Cleft%285%20x%5E%7B2%7D%20-%204%20x%20%2B%202%5Cright%29%20%5Cleft%287%20x%5E%7B2%7D%20%2B%205%20x%20%2B%203%5Cright%29%20" alt="LaTeX: \displaystyle g = \left(5 x^{2} - 4 x + 2\right) \left(7 x^{2} + 5 x + 3\right) " data-equation-content=" \displaystyle g = \left(5 x^{2} - 4 x + 2\right) \left(7 x^{2} + 5 x + 3\right) " /> which also requires the product rule. Pushing down in the new product rule <img class="equation_image" title=" \displaystyle f=5 x^{2} - 4 x + 2 \implies f'=10 x - 4 " src="/equation_images/%20%5Cdisplaystyle%20f%3D5%20x%5E%7B2%7D%20-%204%20x%20%2B%202%20%5Cimplies%20f%27%3D10%20x%20-%204%20" alt="LaTeX: \displaystyle f=5 x^{2} - 4 x + 2 \implies f'=10 x - 4 " data-equation-content=" \displaystyle f=5 x^{2} - 4 x + 2 \implies f'=10 x - 4 " /> and <img class="equation_image" title=" \displaystyle g=7 x^{2} + 5 x + 3 \implies g'=14 x + 5 " src="/equation_images/%20%5Cdisplaystyle%20g%3D7%20x%5E%7B2%7D%20%2B%205%20x%20%2B%203%20%5Cimplies%20g%27%3D14%20x%20%2B%205%20" alt="LaTeX: \displaystyle g=7 x^{2} + 5 x + 3 \implies g'=14 x + 5 " data-equation-content=" \displaystyle g=7 x^{2} + 5 x + 3 \implies g'=14 x + 5 " /> . Popping up a level gives <img class="equation_image" title=" \displaystyle g'=(7 x^{2} + 5 x + 3)(10 x - 4)+(5 x^{2} - 4 x + 2)(14 x + 5) " src="/equation_images/%20%5Cdisplaystyle%20g%27%3D%287%20x%5E%7B2%7D%20%2B%205%20x%20%2B%203%29%2810%20x%20-%204%29%2B%285%20x%5E%7B2%7D%20-%204%20x%20%2B%202%29%2814%20x%20%2B%205%29%20" alt="LaTeX: \displaystyle g'=(7 x^{2} + 5 x + 3)(10 x - 4)+(5 x^{2} - 4 x + 2)(14 x + 5) " data-equation-content=" \displaystyle g'=(7 x^{2} + 5 x + 3)(10 x - 4)+(5 x^{2} - 4 x + 2)(14 x + 5) " /> Popping up again (Back to the original problem) gives <img class="equation_image" title=" \displaystyle f'=(5 x^{2} + 7 x + 2)(\left(10 x - 4\right) \left(7 x^{2} + 5 x + 3\right) + \left(14 x + 5\right) \left(5 x^{2} - 4 x + 2\right))+(\left(5 x^{2} - 4 x + 2\right) \left(7 x^{2} + 5 x + 3\right))(10 x + 7)=\left(10 x - 4\right) \left(5 x^{2} + 7 x + 2\right) \left(7 x^{2} + 5 x + 3\right) + \left(10 x + 7\right) \left(5 x^{2} - 4 x + 2\right) \left(7 x^{2} + 5 x + 3\right) + \left(14 x + 5\right) \left(5 x^{2} - 4 x + 2\right) \left(5 x^{2} + 7 x + 2\right) " src="/equation_images/%20%5Cdisplaystyle%20f%27%3D%285%20x%5E%7B2%7D%20%2B%207%20x%20%2B%202%29%28%5Cleft%2810%20x%20-%204%5Cright%29%20%5Cleft%287%20x%5E%7B2%7D%20%2B%205%20x%20%2B%203%5Cright%29%20%2B%20%5Cleft%2814%20x%20%2B%205%5Cright%29%20%5Cleft%285%20x%5E%7B2%7D%20-%204%20x%20%2B%202%5Cright%29%29%2B%28%5Cleft%285%20x%5E%7B2%7D%20-%204%20x%20%2B%202%5Cright%29%20%5Cleft%287%20x%5E%7B2%7D%20%2B%205%20x%20%2B%203%5Cright%29%29%2810%20x%20%2B%207%29%3D%5Cleft%2810%20x%20-%204%5Cright%29%20%5Cleft%285%20x%5E%7B2%7D%20%2B%207%20x%20%2B%202%5Cright%29%20%5Cleft%287%20x%5E%7B2%7D%20%2B%205%20x%20%2B%203%5Cright%29%20%2B%20%5Cleft%2810%20x%20%2B%207%5Cright%29%20%5Cleft%285%20x%5E%7B2%7D%20-%204%20x%20%2B%202%5Cright%29%20%5Cleft%287%20x%5E%7B2%7D%20%2B%205%20x%20%2B%203%5Cright%29%20%2B%20%5Cleft%2814%20x%20%2B%205%5Cright%29%20%5Cleft%285%20x%5E%7B2%7D%20-%204%20x%20%2B%202%5Cright%29%20%5Cleft%285%20x%5E%7B2%7D%20%2B%207%20x%20%2B%202%5Cright%29%20" alt="LaTeX: \displaystyle f'=(5 x^{2} + 7 x + 2)(\left(10 x - 4\right) \left(7 x^{2} + 5 x + 3\right) + \left(14 x + 5\right) \left(5 x^{2} - 4 x + 2\right))+(\left(5 x^{2} - 4 x + 2\right) \left(7 x^{2} + 5 x + 3\right))(10 x + 7)=\left(10 x - 4\right) \left(5 x^{2} + 7 x + 2\right) \left(7 x^{2} + 5 x + 3\right) + \left(10 x + 7\right) \left(5 x^{2} - 4 x + 2\right) \left(7 x^{2} + 5 x + 3\right) + \left(14 x + 5\right) \left(5 x^{2} - 4 x + 2\right) \left(5 x^{2} + 7 x + 2\right) " data-equation-content=" \displaystyle f'=(5 x^{2} + 7 x + 2)(\left(10 x - 4\right) \left(7 x^{2} + 5 x + 3\right) + \left(14 x + 5\right) \left(5 x^{2} - 4 x + 2\right))+(\left(5 x^{2} - 4 x + 2\right) \left(7 x^{2} + 5 x + 3\right))(10 x + 7)=\left(10 x - 4\right) \left(5 x^{2} + 7 x + 2\right) \left(7 x^{2} + 5 x + 3\right) + \left(10 x + 7\right) \left(5 x^{2} - 4 x + 2\right) \left(7 x^{2} + 5 x + 3\right) + \left(14 x + 5\right) \left(5 x^{2} - 4 x + 2\right) \left(5 x^{2} + 7 x + 2\right) " /> </p> </p>