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Questions: Algebra BusinessCalculus
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Find the derivative of \(\displaystyle y = (- 6 x^{2} + 3 x + 1)(x^{2} + 6 x + 3)(- 8 x^{2} + 8 x + 6)\).
Identifying \(\displaystyle f=- 6 x^{2} + 3 x + 1\) and \(\displaystyle g=\left(- 8 x^{2} + 8 x + 6\right) \left(x^{2} + 6 x + 3\right)\) and using the product rule with \(\displaystyle f=- 6 x^{2} + 3 x + 1 \implies f'=3 - 12 x\). This leaves g as \(\displaystyle g = \left(- 8 x^{2} + 8 x + 6\right) \left(x^{2} + 6 x + 3\right)\) which also requires the product rule. Pushing down in the new product rule \(\displaystyle f=x^{2} + 6 x + 3 \implies f'=2 x + 6\) and \(\displaystyle g=- 8 x^{2} + 8 x + 6 \implies g'=8 - 16 x\). Popping up a level gives \(\displaystyle g'=(- 8 x^{2} + 8 x + 6)(2 x + 6)+(x^{2} + 6 x + 3)(8 - 16 x)\)Popping up again (Back to the original problem) gives \(\displaystyle f'=(- 6 x^{2} + 3 x + 1)(\left(8 - 16 x\right) \left(x^{2} + 6 x + 3\right) + \left(2 x + 6\right) \left(- 8 x^{2} + 8 x + 6\right))+(\left(- 8 x^{2} + 8 x + 6\right) \left(x^{2} + 6 x + 3\right))(3 - 12 x)=\left(3 - 12 x\right) \left(- 8 x^{2} + 8 x + 6\right) \left(x^{2} + 6 x + 3\right) + \left(8 - 16 x\right) \left(- 6 x^{2} + 3 x + 1\right) \left(x^{2} + 6 x + 3\right) + \left(2 x + 6\right) \left(- 8 x^{2} + 8 x + 6\right) \left(- 6 x^{2} + 3 x + 1\right)\)
\begin{question}Find the derivative of $y = (- 6 x^{2} + 3 x + 1)(x^{2} + 6 x + 3)(- 8 x^{2} + 8 x + 6)$.
\soln{9cm}{Identifying $f=- 6 x^{2} + 3 x + 1$ and $g=\left(- 8 x^{2} + 8 x + 6\right) \left(x^{2} + 6 x + 3\right)$ and using the product rule with $f=- 6 x^{2} + 3 x + 1 \implies f'=3 - 12 x$. This leaves g as $g = \left(- 8 x^{2} + 8 x + 6\right) \left(x^{2} + 6 x + 3\right)$ which also requires the product rule. Pushing down in the new product rule $f=x^{2} + 6 x + 3 \implies f'=2 x + 6$ and $g=- 8 x^{2} + 8 x + 6 \implies g'=8 - 16 x$. Popping up a level gives $g'=(- 8 x^{2} + 8 x + 6)(2 x + 6)+(x^{2} + 6 x + 3)(8 - 16 x)$Popping up again (Back to the original problem) gives $f'=(- 6 x^{2} + 3 x + 1)(\left(8 - 16 x\right) \left(x^{2} + 6 x + 3\right) + \left(2 x + 6\right) \left(- 8 x^{2} + 8 x + 6\right))+(\left(- 8 x^{2} + 8 x + 6\right) \left(x^{2} + 6 x + 3\right))(3 - 12 x)=\left(3 - 12 x\right) \left(- 8 x^{2} + 8 x + 6\right) \left(x^{2} + 6 x + 3\right) + \left(8 - 16 x\right) \left(- 6 x^{2} + 3 x + 1\right) \left(x^{2} + 6 x + 3\right) + \left(2 x + 6\right) \left(- 8 x^{2} + 8 x + 6\right) \left(- 6 x^{2} + 3 x + 1\right)$}
\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 = (- 6 x^{2} + 3 x + 1)(x^{2} + 6 x + 3)(- 8 x^{2} + 8 x + 6) " src="/equation_images/%20%5Cdisplaystyle%20y%20%3D%20%28-%206%20x%5E%7B2%7D%20%2B%203%20x%20%2B%201%29%28x%5E%7B2%7D%20%2B%206%20x%20%2B%203%29%28-%208%20x%5E%7B2%7D%20%2B%208%20x%20%2B%206%29%20" alt="LaTeX: \displaystyle y = (- 6 x^{2} + 3 x + 1)(x^{2} + 6 x + 3)(- 8 x^{2} + 8 x + 6) " data-equation-content=" \displaystyle y = (- 6 x^{2} + 3 x + 1)(x^{2} + 6 x + 3)(- 8 x^{2} + 8 x + 6) " /> .</p> </p><p> <p>Identifying <img class="equation_image" title=" \displaystyle f=- 6 x^{2} + 3 x + 1 " src="/equation_images/%20%5Cdisplaystyle%20f%3D-%206%20x%5E%7B2%7D%20%2B%203%20x%20%2B%201%20" alt="LaTeX: \displaystyle f=- 6 x^{2} + 3 x + 1 " data-equation-content=" \displaystyle f=- 6 x^{2} + 3 x + 1 " /> and <img class="equation_image" title=" \displaystyle g=\left(- 8 x^{2} + 8 x + 6\right) \left(x^{2} + 6 x + 3\right) " src="/equation_images/%20%5Cdisplaystyle%20g%3D%5Cleft%28-%208%20x%5E%7B2%7D%20%2B%208%20x%20%2B%206%5Cright%29%20%5Cleft%28x%5E%7B2%7D%20%2B%206%20x%20%2B%203%5Cright%29%20" alt="LaTeX: \displaystyle g=\left(- 8 x^{2} + 8 x + 6\right) \left(x^{2} + 6 x + 3\right) " data-equation-content=" \displaystyle g=\left(- 8 x^{2} + 8 x + 6\right) \left(x^{2} + 6 x + 3\right) " /> and using the product rule with <img class="equation_image" title=" \displaystyle f=- 6 x^{2} + 3 x + 1 \implies f'=3 - 12 x " src="/equation_images/%20%5Cdisplaystyle%20f%3D-%206%20x%5E%7B2%7D%20%2B%203%20x%20%2B%201%20%5Cimplies%20f%27%3D3%20-%2012%20x%20" alt="LaTeX: \displaystyle f=- 6 x^{2} + 3 x + 1 \implies f'=3 - 12 x " data-equation-content=" \displaystyle f=- 6 x^{2} + 3 x + 1 \implies f'=3 - 12 x " /> . This leaves g as <img class="equation_image" title=" \displaystyle g = \left(- 8 x^{2} + 8 x + 6\right) \left(x^{2} + 6 x + 3\right) " src="/equation_images/%20%5Cdisplaystyle%20g%20%3D%20%5Cleft%28-%208%20x%5E%7B2%7D%20%2B%208%20x%20%2B%206%5Cright%29%20%5Cleft%28x%5E%7B2%7D%20%2B%206%20x%20%2B%203%5Cright%29%20" alt="LaTeX: \displaystyle g = \left(- 8 x^{2} + 8 x + 6\right) \left(x^{2} + 6 x + 3\right) " data-equation-content=" \displaystyle g = \left(- 8 x^{2} + 8 x + 6\right) \left(x^{2} + 6 x + 3\right) " /> which also requires the product rule. Pushing down in the new product rule <img class="equation_image" title=" \displaystyle f=x^{2} + 6 x + 3 \implies f'=2 x + 6 " src="/equation_images/%20%5Cdisplaystyle%20f%3Dx%5E%7B2%7D%20%2B%206%20x%20%2B%203%20%5Cimplies%20f%27%3D2%20x%20%2B%206%20" alt="LaTeX: \displaystyle f=x^{2} + 6 x + 3 \implies f'=2 x + 6 " data-equation-content=" \displaystyle f=x^{2} + 6 x + 3 \implies f'=2 x + 6 " /> and <img class="equation_image" title=" \displaystyle g=- 8 x^{2} + 8 x + 6 \implies g'=8 - 16 x " src="/equation_images/%20%5Cdisplaystyle%20g%3D-%208%20x%5E%7B2%7D%20%2B%208%20x%20%2B%206%20%5Cimplies%20g%27%3D8%20-%2016%20x%20" alt="LaTeX: \displaystyle g=- 8 x^{2} + 8 x + 6 \implies g'=8 - 16 x " data-equation-content=" \displaystyle g=- 8 x^{2} + 8 x + 6 \implies g'=8 - 16 x " /> . Popping up a level gives <img class="equation_image" title=" \displaystyle g'=(- 8 x^{2} + 8 x + 6)(2 x + 6)+(x^{2} + 6 x + 3)(8 - 16 x) " src="/equation_images/%20%5Cdisplaystyle%20g%27%3D%28-%208%20x%5E%7B2%7D%20%2B%208%20x%20%2B%206%29%282%20x%20%2B%206%29%2B%28x%5E%7B2%7D%20%2B%206%20x%20%2B%203%29%288%20-%2016%20x%29%20" alt="LaTeX: \displaystyle g'=(- 8 x^{2} + 8 x + 6)(2 x + 6)+(x^{2} + 6 x + 3)(8 - 16 x) " data-equation-content=" \displaystyle g'=(- 8 x^{2} + 8 x + 6)(2 x + 6)+(x^{2} + 6 x + 3)(8 - 16 x) " /> Popping up again (Back to the original problem) gives <img class="equation_image" title=" \displaystyle f'=(- 6 x^{2} + 3 x + 1)(\left(8 - 16 x\right) \left(x^{2} + 6 x + 3\right) + \left(2 x + 6\right) \left(- 8 x^{2} + 8 x + 6\right))+(\left(- 8 x^{2} + 8 x + 6\right) \left(x^{2} + 6 x + 3\right))(3 - 12 x)=\left(3 - 12 x\right) \left(- 8 x^{2} + 8 x + 6\right) \left(x^{2} + 6 x + 3\right) + \left(8 - 16 x\right) \left(- 6 x^{2} + 3 x + 1\right) \left(x^{2} + 6 x + 3\right) + \left(2 x + 6\right) \left(- 8 x^{2} + 8 x + 6\right) \left(- 6 x^{2} + 3 x + 1\right) " src="/equation_images/%20%5Cdisplaystyle%20f%27%3D%28-%206%20x%5E%7B2%7D%20%2B%203%20x%20%2B%201%29%28%5Cleft%288%20-%2016%20x%5Cright%29%20%5Cleft%28x%5E%7B2%7D%20%2B%206%20x%20%2B%203%5Cright%29%20%2B%20%5Cleft%282%20x%20%2B%206%5Cright%29%20%5Cleft%28-%208%20x%5E%7B2%7D%20%2B%208%20x%20%2B%206%5Cright%29%29%2B%28%5Cleft%28-%208%20x%5E%7B2%7D%20%2B%208%20x%20%2B%206%5Cright%29%20%5Cleft%28x%5E%7B2%7D%20%2B%206%20x%20%2B%203%5Cright%29%29%283%20-%2012%20x%29%3D%5Cleft%283%20-%2012%20x%5Cright%29%20%5Cleft%28-%208%20x%5E%7B2%7D%20%2B%208%20x%20%2B%206%5Cright%29%20%5Cleft%28x%5E%7B2%7D%20%2B%206%20x%20%2B%203%5Cright%29%20%2B%20%5Cleft%288%20-%2016%20x%5Cright%29%20%5Cleft%28-%206%20x%5E%7B2%7D%20%2B%203%20x%20%2B%201%5Cright%29%20%5Cleft%28x%5E%7B2%7D%20%2B%206%20x%20%2B%203%5Cright%29%20%2B%20%5Cleft%282%20x%20%2B%206%5Cright%29%20%5Cleft%28-%208%20x%5E%7B2%7D%20%2B%208%20x%20%2B%206%5Cright%29%20%5Cleft%28-%206%20x%5E%7B2%7D%20%2B%203%20x%20%2B%201%5Cright%29%20" alt="LaTeX: \displaystyle f'=(- 6 x^{2} + 3 x + 1)(\left(8 - 16 x\right) \left(x^{2} + 6 x + 3\right) + \left(2 x + 6\right) \left(- 8 x^{2} + 8 x + 6\right))+(\left(- 8 x^{2} + 8 x + 6\right) \left(x^{2} + 6 x + 3\right))(3 - 12 x)=\left(3 - 12 x\right) \left(- 8 x^{2} + 8 x + 6\right) \left(x^{2} + 6 x + 3\right) + \left(8 - 16 x\right) \left(- 6 x^{2} + 3 x + 1\right) \left(x^{2} + 6 x + 3\right) + \left(2 x + 6\right) \left(- 8 x^{2} + 8 x + 6\right) \left(- 6 x^{2} + 3 x + 1\right) " data-equation-content=" \displaystyle f'=(- 6 x^{2} + 3 x + 1)(\left(8 - 16 x\right) \left(x^{2} + 6 x + 3\right) + \left(2 x + 6\right) \left(- 8 x^{2} + 8 x + 6\right))+(\left(- 8 x^{2} + 8 x + 6\right) \left(x^{2} + 6 x + 3\right))(3 - 12 x)=\left(3 - 12 x\right) \left(- 8 x^{2} + 8 x + 6\right) \left(x^{2} + 6 x + 3\right) + \left(8 - 16 x\right) \left(- 6 x^{2} + 3 x + 1\right) \left(x^{2} + 6 x + 3\right) + \left(2 x + 6\right) \left(- 8 x^{2} + 8 x + 6\right) \left(- 6 x^{2} + 3 x + 1\right) " /> </p> </p>