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Questions: Algebra BusinessCalculus
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Find the derivative of \(\displaystyle y = (9 x^{2} + 9 x + 9)(\log{\left(x \right)})(6 x^{2} - 3 x + 6)\).
Identifying \(\displaystyle f=9 x^{2} + 9 x + 9\) and \(\displaystyle g=\left(6 x^{2} - 3 x + 6\right) \log{\left(x \right)}\) and using the product rule with \(\displaystyle f=9 x^{2} + 9 x + 9 \implies f'=18 x + 9\). This leaves g as \(\displaystyle g = \left(6 x^{2} - 3 x + 6\right) \log{\left(x \right)}\) which also requires the product rule. Pushing down in the new product rule \(\displaystyle f=\log{\left(x \right)} \implies f'=\frac{1}{x}\) and \(\displaystyle g=6 x^{2} - 3 x + 6 \implies g'=12 x - 3\). Popping up a level gives \(\displaystyle g'=(6 x^{2} - 3 x + 6)(\frac{1}{x})+(\log{\left(x \right)})(12 x - 3)\)Popping up again (Back to the original problem) gives \(\displaystyle f'=(9 x^{2} + 9 x + 9)(\left(12 x - 3\right) \log{\left(x \right)} + \frac{6 x^{2} - 3 x + 6}{x})+(\left(6 x^{2} - 3 x + 6\right) \log{\left(x \right)})(18 x + 9)=\left(12 x - 3\right) \left(9 x^{2} + 9 x + 9\right) \log{\left(x \right)} + \left(18 x + 9\right) \left(6 x^{2} - 3 x + 6\right) \log{\left(x \right)} + \frac{\left(6 x^{2} - 3 x + 6\right) \left(9 x^{2} + 9 x + 9\right)}{x}\)
\begin{question}Find the derivative of $y = (9 x^{2} + 9 x + 9)(\log{\left(x \right)})(6 x^{2} - 3 x + 6)$.
\soln{9cm}{Identifying $f=9 x^{2} + 9 x + 9$ and $g=\left(6 x^{2} - 3 x + 6\right) \log{\left(x \right)}$ and using the product rule with $f=9 x^{2} + 9 x + 9 \implies f'=18 x + 9$. This leaves g as $g = \left(6 x^{2} - 3 x + 6\right) \log{\left(x \right)}$ which also requires the product rule. Pushing down in the new product rule $f=\log{\left(x \right)} \implies f'=\frac{1}{x}$ and $g=6 x^{2} - 3 x + 6 \implies g'=12 x - 3$. Popping up a level gives $g'=(6 x^{2} - 3 x + 6)(\frac{1}{x})+(\log{\left(x \right)})(12 x - 3)$Popping up again (Back to the original problem) gives $f'=(9 x^{2} + 9 x + 9)(\left(12 x - 3\right) \log{\left(x \right)} + \frac{6 x^{2} - 3 x + 6}{x})+(\left(6 x^{2} - 3 x + 6\right) \log{\left(x \right)})(18 x + 9)=\left(12 x - 3\right) \left(9 x^{2} + 9 x + 9\right) \log{\left(x \right)} + \left(18 x + 9\right) \left(6 x^{2} - 3 x + 6\right) \log{\left(x \right)} + \frac{\left(6 x^{2} - 3 x + 6\right) \left(9 x^{2} + 9 x + 9\right)}{x}$}
\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 = (9 x^{2} + 9 x + 9)(\log{\left(x \right)})(6 x^{2} - 3 x + 6) " src="/equation_images/%20%5Cdisplaystyle%20y%20%3D%20%289%20x%5E%7B2%7D%20%2B%209%20x%20%2B%209%29%28%5Clog%7B%5Cleft%28x%20%5Cright%29%7D%29%286%20x%5E%7B2%7D%20-%203%20x%20%2B%206%29%20" alt="LaTeX: \displaystyle y = (9 x^{2} + 9 x + 9)(\log{\left(x \right)})(6 x^{2} - 3 x + 6) " data-equation-content=" \displaystyle y = (9 x^{2} + 9 x + 9)(\log{\left(x \right)})(6 x^{2} - 3 x + 6) " /> .</p> </p><p> <p>Identifying <img class="equation_image" title=" \displaystyle f=9 x^{2} + 9 x + 9 " src="/equation_images/%20%5Cdisplaystyle%20f%3D9%20x%5E%7B2%7D%20%2B%209%20x%20%2B%209%20" alt="LaTeX: \displaystyle f=9 x^{2} + 9 x + 9 " data-equation-content=" \displaystyle f=9 x^{2} + 9 x + 9 " /> and <img class="equation_image" title=" \displaystyle g=\left(6 x^{2} - 3 x + 6\right) \log{\left(x \right)} " src="/equation_images/%20%5Cdisplaystyle%20g%3D%5Cleft%286%20x%5E%7B2%7D%20-%203%20x%20%2B%206%5Cright%29%20%5Clog%7B%5Cleft%28x%20%5Cright%29%7D%20" alt="LaTeX: \displaystyle g=\left(6 x^{2} - 3 x + 6\right) \log{\left(x \right)} " data-equation-content=" \displaystyle g=\left(6 x^{2} - 3 x + 6\right) \log{\left(x \right)} " /> and using the product rule with <img class="equation_image" title=" \displaystyle f=9 x^{2} + 9 x + 9 \implies f'=18 x + 9 " src="/equation_images/%20%5Cdisplaystyle%20f%3D9%20x%5E%7B2%7D%20%2B%209%20x%20%2B%209%20%5Cimplies%20f%27%3D18%20x%20%2B%209%20" alt="LaTeX: \displaystyle f=9 x^{2} + 9 x + 9 \implies f'=18 x + 9 " data-equation-content=" \displaystyle f=9 x^{2} + 9 x + 9 \implies f'=18 x + 9 " /> . This leaves g as <img class="equation_image" title=" \displaystyle g = \left(6 x^{2} - 3 x + 6\right) \log{\left(x \right)} " src="/equation_images/%20%5Cdisplaystyle%20g%20%3D%20%5Cleft%286%20x%5E%7B2%7D%20-%203%20x%20%2B%206%5Cright%29%20%5Clog%7B%5Cleft%28x%20%5Cright%29%7D%20" alt="LaTeX: \displaystyle g = \left(6 x^{2} - 3 x + 6\right) \log{\left(x \right)} " data-equation-content=" \displaystyle g = \left(6 x^{2} - 3 x + 6\right) \log{\left(x \right)} " /> which also requires the product rule. Pushing down in the new product rule <img class="equation_image" title=" \displaystyle f=\log{\left(x \right)} \implies f'=\frac{1}{x} " src="/equation_images/%20%5Cdisplaystyle%20f%3D%5Clog%7B%5Cleft%28x%20%5Cright%29%7D%20%5Cimplies%20f%27%3D%5Cfrac%7B1%7D%7Bx%7D%20" alt="LaTeX: \displaystyle f=\log{\left(x \right)} \implies f'=\frac{1}{x} " data-equation-content=" \displaystyle f=\log{\left(x \right)} \implies f'=\frac{1}{x} " /> and <img class="equation_image" title=" \displaystyle g=6 x^{2} - 3 x + 6 \implies g'=12 x - 3 " src="/equation_images/%20%5Cdisplaystyle%20g%3D6%20x%5E%7B2%7D%20-%203%20x%20%2B%206%20%5Cimplies%20g%27%3D12%20x%20-%203%20" alt="LaTeX: \displaystyle g=6 x^{2} - 3 x + 6 \implies g'=12 x - 3 " data-equation-content=" \displaystyle g=6 x^{2} - 3 x + 6 \implies g'=12 x - 3 " /> . Popping up a level gives <img class="equation_image" title=" \displaystyle g'=(6 x^{2} - 3 x + 6)(\frac{1}{x})+(\log{\left(x \right)})(12 x - 3) " src="/equation_images/%20%5Cdisplaystyle%20g%27%3D%286%20x%5E%7B2%7D%20-%203%20x%20%2B%206%29%28%5Cfrac%7B1%7D%7Bx%7D%29%2B%28%5Clog%7B%5Cleft%28x%20%5Cright%29%7D%29%2812%20x%20-%203%29%20" alt="LaTeX: \displaystyle g'=(6 x^{2} - 3 x + 6)(\frac{1}{x})+(\log{\left(x \right)})(12 x - 3) " data-equation-content=" \displaystyle g'=(6 x^{2} - 3 x + 6)(\frac{1}{x})+(\log{\left(x \right)})(12 x - 3) " /> Popping up again (Back to the original problem) gives <img class="equation_image" title=" \displaystyle f'=(9 x^{2} + 9 x + 9)(\left(12 x - 3\right) \log{\left(x \right)} + \frac{6 x^{2} - 3 x + 6}{x})+(\left(6 x^{2} - 3 x + 6\right) \log{\left(x \right)})(18 x + 9)=\left(12 x - 3\right) \left(9 x^{2} + 9 x + 9\right) \log{\left(x \right)} + \left(18 x + 9\right) \left(6 x^{2} - 3 x + 6\right) \log{\left(x \right)} + \frac{\left(6 x^{2} - 3 x + 6\right) \left(9 x^{2} + 9 x + 9\right)}{x} " src="/equation_images/%20%5Cdisplaystyle%20f%27%3D%289%20x%5E%7B2%7D%20%2B%209%20x%20%2B%209%29%28%5Cleft%2812%20x%20-%203%5Cright%29%20%5Clog%7B%5Cleft%28x%20%5Cright%29%7D%20%2B%20%5Cfrac%7B6%20x%5E%7B2%7D%20-%203%20x%20%2B%206%7D%7Bx%7D%29%2B%28%5Cleft%286%20x%5E%7B2%7D%20-%203%20x%20%2B%206%5Cright%29%20%5Clog%7B%5Cleft%28x%20%5Cright%29%7D%29%2818%20x%20%2B%209%29%3D%5Cleft%2812%20x%20-%203%5Cright%29%20%5Cleft%289%20x%5E%7B2%7D%20%2B%209%20x%20%2B%209%5Cright%29%20%5Clog%7B%5Cleft%28x%20%5Cright%29%7D%20%2B%20%5Cleft%2818%20x%20%2B%209%5Cright%29%20%5Cleft%286%20x%5E%7B2%7D%20-%203%20x%20%2B%206%5Cright%29%20%5Clog%7B%5Cleft%28x%20%5Cright%29%7D%20%2B%20%5Cfrac%7B%5Cleft%286%20x%5E%7B2%7D%20-%203%20x%20%2B%206%5Cright%29%20%5Cleft%289%20x%5E%7B2%7D%20%2B%209%20x%20%2B%209%5Cright%29%7D%7Bx%7D%20" alt="LaTeX: \displaystyle f'=(9 x^{2} + 9 x + 9)(\left(12 x - 3\right) \log{\left(x \right)} + \frac{6 x^{2} - 3 x + 6}{x})+(\left(6 x^{2} - 3 x + 6\right) \log{\left(x \right)})(18 x + 9)=\left(12 x - 3\right) \left(9 x^{2} + 9 x + 9\right) \log{\left(x \right)} + \left(18 x + 9\right) \left(6 x^{2} - 3 x + 6\right) \log{\left(x \right)} + \frac{\left(6 x^{2} - 3 x + 6\right) \left(9 x^{2} + 9 x + 9\right)}{x} " data-equation-content=" \displaystyle f'=(9 x^{2} + 9 x + 9)(\left(12 x - 3\right) \log{\left(x \right)} + \frac{6 x^{2} - 3 x + 6}{x})+(\left(6 x^{2} - 3 x + 6\right) \log{\left(x \right)})(18 x + 9)=\left(12 x - 3\right) \left(9 x^{2} + 9 x + 9\right) \log{\left(x \right)} + \left(18 x + 9\right) \left(6 x^{2} - 3 x + 6\right) \log{\left(x \right)} + \frac{\left(6 x^{2} - 3 x + 6\right) \left(9 x^{2} + 9 x + 9\right)}{x} " /> </p> </p>