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Questions: Algebra BusinessCalculus
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Find the derivative of \(\displaystyle y = (- 4 x - 1)(8 - 9 x)(9 - 5 x)\).
Identifying \(\displaystyle f=- 4 x - 1\) and \(\displaystyle g=\left(8 - 9 x\right) \left(9 - 5 x\right)\) and using the product rule with \(\displaystyle f=- 4 x - 1 \implies f'=-4\). This leaves g as \(\displaystyle g = \left(8 - 9 x\right) \left(9 - 5 x\right)\) which also requires the product rule. Pushing down in the new product rule \(\displaystyle f=8 - 9 x \implies f'=-9\) and \(\displaystyle g=9 - 5 x \implies g'=-5\). Popping up a level gives \(\displaystyle g'=(9 - 5 x)(-9)+(8 - 9 x)(-5)\)Popping up again (Back to the original problem) gives \(\displaystyle f'=(- 4 x - 1)(90 x - 121)+(\left(8 - 9 x\right) \left(9 - 5 x\right))(-4)=- 5 \left(8 - 9 x\right) \left(- 4 x - 1\right) + \left(9 - 5 x\right) \left(36 x - 32\right) + \left(9 - 5 x\right) \left(36 x + 9\right)\)
\begin{question}Find the derivative of $y = (- 4 x - 1)(8 - 9 x)(9 - 5 x)$.
\soln{9cm}{Identifying $f=- 4 x - 1$ and $g=\left(8 - 9 x\right) \left(9 - 5 x\right)$ and using the product rule with $f=- 4 x - 1 \implies f'=-4$. This leaves g as $g = \left(8 - 9 x\right) \left(9 - 5 x\right)$ which also requires the product rule. Pushing down in the new product rule $f=8 - 9 x \implies f'=-9$ and $g=9 - 5 x \implies g'=-5$. Popping up a level gives $g'=(9 - 5 x)(-9)+(8 - 9 x)(-5)$Popping up again (Back to the original problem) gives $f'=(- 4 x - 1)(90 x - 121)+(\left(8 - 9 x\right) \left(9 - 5 x\right))(-4)=- 5 \left(8 - 9 x\right) \left(- 4 x - 1\right) + \left(9 - 5 x\right) \left(36 x - 32\right) + \left(9 - 5 x\right) \left(36 x + 9\right)$}
\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 = (- 4 x - 1)(8 - 9 x)(9 - 5 x) " src="/equation_images/%20%5Cdisplaystyle%20y%20%3D%20%28-%204%20x%20-%201%29%288%20-%209%20x%29%289%20-%205%20x%29%20" alt="LaTeX: \displaystyle y = (- 4 x - 1)(8 - 9 x)(9 - 5 x) " data-equation-content=" \displaystyle y = (- 4 x - 1)(8 - 9 x)(9 - 5 x) " /> .</p> </p>
<p> <p>Identifying <img class="equation_image" title=" \displaystyle f=- 4 x - 1 " src="/equation_images/%20%5Cdisplaystyle%20f%3D-%204%20x%20-%201%20" alt="LaTeX: \displaystyle f=- 4 x - 1 " data-equation-content=" \displaystyle f=- 4 x - 1 " /> and <img class="equation_image" title=" \displaystyle g=\left(8 - 9 x\right) \left(9 - 5 x\right) " src="/equation_images/%20%5Cdisplaystyle%20g%3D%5Cleft%288%20-%209%20x%5Cright%29%20%5Cleft%289%20-%205%20x%5Cright%29%20" alt="LaTeX: \displaystyle g=\left(8 - 9 x\right) \left(9 - 5 x\right) " data-equation-content=" \displaystyle g=\left(8 - 9 x\right) \left(9 - 5 x\right) " /> and using the product rule with <img class="equation_image" title=" \displaystyle f=- 4 x - 1 \implies f'=-4 " src="/equation_images/%20%5Cdisplaystyle%20f%3D-%204%20x%20-%201%20%5Cimplies%20f%27%3D-4%20" alt="LaTeX: \displaystyle f=- 4 x - 1 \implies f'=-4 " data-equation-content=" \displaystyle f=- 4 x - 1 \implies f'=-4 " /> . This leaves g as <img class="equation_image" title=" \displaystyle g = \left(8 - 9 x\right) \left(9 - 5 x\right) " src="/equation_images/%20%5Cdisplaystyle%20g%20%3D%20%5Cleft%288%20-%209%20x%5Cright%29%20%5Cleft%289%20-%205%20x%5Cright%29%20" alt="LaTeX: \displaystyle g = \left(8 - 9 x\right) \left(9 - 5 x\right) " data-equation-content=" \displaystyle g = \left(8 - 9 x\right) \left(9 - 5 x\right) " /> which also requires the product rule. Pushing down in the new product rule <img class="equation_image" title=" \displaystyle f=8 - 9 x \implies f'=-9 " src="/equation_images/%20%5Cdisplaystyle%20f%3D8%20-%209%20x%20%5Cimplies%20f%27%3D-9%20" alt="LaTeX: \displaystyle f=8 - 9 x \implies f'=-9 " data-equation-content=" \displaystyle f=8 - 9 x \implies f'=-9 " /> and <img class="equation_image" title=" \displaystyle g=9 - 5 x \implies g'=-5 " src="/equation_images/%20%5Cdisplaystyle%20g%3D9%20-%205%20x%20%5Cimplies%20g%27%3D-5%20" alt="LaTeX: \displaystyle g=9 - 5 x \implies g'=-5 " data-equation-content=" \displaystyle g=9 - 5 x \implies g'=-5 " /> . Popping up a level gives <img class="equation_image" title=" \displaystyle g'=(9 - 5 x)(-9)+(8 - 9 x)(-5) " src="/equation_images/%20%5Cdisplaystyle%20g%27%3D%289%20-%205%20x%29%28-9%29%2B%288%20-%209%20x%29%28-5%29%20" alt="LaTeX: \displaystyle g'=(9 - 5 x)(-9)+(8 - 9 x)(-5) " data-equation-content=" \displaystyle g'=(9 - 5 x)(-9)+(8 - 9 x)(-5) " /> Popping up again (Back to the original problem) gives <img class="equation_image" title=" \displaystyle f'=(- 4 x - 1)(90 x - 121)+(\left(8 - 9 x\right) \left(9 - 5 x\right))(-4)=- 5 \left(8 - 9 x\right) \left(- 4 x - 1\right) + \left(9 - 5 x\right) \left(36 x - 32\right) + \left(9 - 5 x\right) \left(36 x + 9\right) " src="/equation_images/%20%5Cdisplaystyle%20f%27%3D%28-%204%20x%20-%201%29%2890%20x%20-%20121%29%2B%28%5Cleft%288%20-%209%20x%5Cright%29%20%5Cleft%289%20-%205%20x%5Cright%29%29%28-4%29%3D-%205%20%5Cleft%288%20-%209%20x%5Cright%29%20%5Cleft%28-%204%20x%20-%201%5Cright%29%20%2B%20%5Cleft%289%20-%205%20x%5Cright%29%20%5Cleft%2836%20x%20-%2032%5Cright%29%20%2B%20%5Cleft%289%20-%205%20x%5Cright%29%20%5Cleft%2836%20x%20%2B%209%5Cright%29%20" alt="LaTeX: \displaystyle f'=(- 4 x - 1)(90 x - 121)+(\left(8 - 9 x\right) \left(9 - 5 x\right))(-4)=- 5 \left(8 - 9 x\right) \left(- 4 x - 1\right) + \left(9 - 5 x\right) \left(36 x - 32\right) + \left(9 - 5 x\right) \left(36 x + 9\right) " data-equation-content=" \displaystyle f'=(- 4 x - 1)(90 x - 121)+(\left(8 - 9 x\right) \left(9 - 5 x\right))(-4)=- 5 \left(8 - 9 x\right) \left(- 4 x - 1\right) + \left(9 - 5 x\right) \left(36 x - 32\right) + \left(9 - 5 x\right) \left(36 x + 9\right) " /> </p> </p>