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Questions: Algebra BusinessCalculus
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Find the derivative of \(\displaystyle y = (4 - 2 x)(9 x - 7)(- 5 x - 4)\).
Identifying \(\displaystyle f=4 - 2 x\) and \(\displaystyle g=\left(- 5 x - 4\right) \left(9 x - 7\right)\) and using the product rule with \(\displaystyle f=4 - 2 x \implies f'=-2\). This leaves g as \(\displaystyle g = \left(- 5 x - 4\right) \left(9 x - 7\right)\) which also requires the product rule. Pushing down in the new product rule \(\displaystyle f=9 x - 7 \implies f'=9\) and \(\displaystyle g=- 5 x - 4 \implies g'=-5\). Popping up a level gives \(\displaystyle g'=(- 5 x - 4)(9)+(9 x - 7)(-5)\)Popping up again (Back to the original problem) gives \(\displaystyle f'=(4 - 2 x)(- 90 x - 1)+(\left(- 5 x - 4\right) \left(9 x - 7\right))(-2)=\left(4 - 2 x\right) \left(35 - 45 x\right) + \left(4 - 2 x\right) \left(- 45 x - 36\right) - 2 \left(- 5 x - 4\right) \left(9 x - 7\right)\)
\begin{question}Find the derivative of $y = (4 - 2 x)(9 x - 7)(- 5 x - 4)$.
\soln{9cm}{Identifying $f=4 - 2 x$ and $g=\left(- 5 x - 4\right) \left(9 x - 7\right)$ and using the product rule with $f=4 - 2 x \implies f'=-2$. This leaves g as $g = \left(- 5 x - 4\right) \left(9 x - 7\right)$ which also requires the product rule. Pushing down in the new product rule $f=9 x - 7 \implies f'=9$ and $g=- 5 x - 4 \implies g'=-5$. Popping up a level gives $g'=(- 5 x - 4)(9)+(9 x - 7)(-5)$Popping up again (Back to the original problem) gives $f'=(4 - 2 x)(- 90 x - 1)+(\left(- 5 x - 4\right) \left(9 x - 7\right))(-2)=\left(4 - 2 x\right) \left(35 - 45 x\right) + \left(4 - 2 x\right) \left(- 45 x - 36\right) - 2 \left(- 5 x - 4\right) \left(9 x - 7\right)$}
\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 - 2 x)(9 x - 7)(- 5 x - 4) " src="/equation_images/%20%5Cdisplaystyle%20y%20%3D%20%284%20-%202%20x%29%289%20x%20-%207%29%28-%205%20x%20-%204%29%20" alt="LaTeX: \displaystyle y = (4 - 2 x)(9 x - 7)(- 5 x - 4) " data-equation-content=" \displaystyle y = (4 - 2 x)(9 x - 7)(- 5 x - 4) " /> .</p> </p>
<p> <p>Identifying <img class="equation_image" title=" \displaystyle f=4 - 2 x " src="/equation_images/%20%5Cdisplaystyle%20f%3D4%20-%202%20x%20" alt="LaTeX: \displaystyle f=4 - 2 x " data-equation-content=" \displaystyle f=4 - 2 x " /> and <img class="equation_image" title=" \displaystyle g=\left(- 5 x - 4\right) \left(9 x - 7\right) " src="/equation_images/%20%5Cdisplaystyle%20g%3D%5Cleft%28-%205%20x%20-%204%5Cright%29%20%5Cleft%289%20x%20-%207%5Cright%29%20" alt="LaTeX: \displaystyle g=\left(- 5 x - 4\right) \left(9 x - 7\right) " data-equation-content=" \displaystyle g=\left(- 5 x - 4\right) \left(9 x - 7\right) " /> and using the product rule with <img class="equation_image" title=" \displaystyle f=4 - 2 x \implies f'=-2 " src="/equation_images/%20%5Cdisplaystyle%20f%3D4%20-%202%20x%20%5Cimplies%20f%27%3D-2%20" alt="LaTeX: \displaystyle f=4 - 2 x \implies f'=-2 " data-equation-content=" \displaystyle f=4 - 2 x \implies f'=-2 " /> . This leaves g as <img class="equation_image" title=" \displaystyle g = \left(- 5 x - 4\right) \left(9 x - 7\right) " src="/equation_images/%20%5Cdisplaystyle%20g%20%3D%20%5Cleft%28-%205%20x%20-%204%5Cright%29%20%5Cleft%289%20x%20-%207%5Cright%29%20" alt="LaTeX: \displaystyle g = \left(- 5 x - 4\right) \left(9 x - 7\right) " data-equation-content=" \displaystyle g = \left(- 5 x - 4\right) \left(9 x - 7\right) " /> which also requires the product rule. Pushing down in the new product rule <img class="equation_image" title=" \displaystyle f=9 x - 7 \implies f'=9 " src="/equation_images/%20%5Cdisplaystyle%20f%3D9%20x%20-%207%20%5Cimplies%20f%27%3D9%20" alt="LaTeX: \displaystyle f=9 x - 7 \implies f'=9 " data-equation-content=" \displaystyle f=9 x - 7 \implies f'=9 " /> and <img class="equation_image" title=" \displaystyle g=- 5 x - 4 \implies g'=-5 " src="/equation_images/%20%5Cdisplaystyle%20g%3D-%205%20x%20-%204%20%5Cimplies%20g%27%3D-5%20" alt="LaTeX: \displaystyle g=- 5 x - 4 \implies g'=-5 " data-equation-content=" \displaystyle g=- 5 x - 4 \implies g'=-5 " /> . Popping up a level gives <img class="equation_image" title=" \displaystyle g'=(- 5 x - 4)(9)+(9 x - 7)(-5) " src="/equation_images/%20%5Cdisplaystyle%20g%27%3D%28-%205%20x%20-%204%29%289%29%2B%289%20x%20-%207%29%28-5%29%20" alt="LaTeX: \displaystyle g'=(- 5 x - 4)(9)+(9 x - 7)(-5) " data-equation-content=" \displaystyle g'=(- 5 x - 4)(9)+(9 x - 7)(-5) " /> Popping up again (Back to the original problem) gives <img class="equation_image" title=" \displaystyle f'=(4 - 2 x)(- 90 x - 1)+(\left(- 5 x - 4\right) \left(9 x - 7\right))(-2)=\left(4 - 2 x\right) \left(35 - 45 x\right) + \left(4 - 2 x\right) \left(- 45 x - 36\right) - 2 \left(- 5 x - 4\right) \left(9 x - 7\right) " src="/equation_images/%20%5Cdisplaystyle%20f%27%3D%284%20-%202%20x%29%28-%2090%20x%20-%201%29%2B%28%5Cleft%28-%205%20x%20-%204%5Cright%29%20%5Cleft%289%20x%20-%207%5Cright%29%29%28-2%29%3D%5Cleft%284%20-%202%20x%5Cright%29%20%5Cleft%2835%20-%2045%20x%5Cright%29%20%2B%20%5Cleft%284%20-%202%20x%5Cright%29%20%5Cleft%28-%2045%20x%20-%2036%5Cright%29%20-%202%20%5Cleft%28-%205%20x%20-%204%5Cright%29%20%5Cleft%289%20x%20-%207%5Cright%29%20" alt="LaTeX: \displaystyle f'=(4 - 2 x)(- 90 x - 1)+(\left(- 5 x - 4\right) \left(9 x - 7\right))(-2)=\left(4 - 2 x\right) \left(35 - 45 x\right) + \left(4 - 2 x\right) \left(- 45 x - 36\right) - 2 \left(- 5 x - 4\right) \left(9 x - 7\right) " data-equation-content=" \displaystyle f'=(4 - 2 x)(- 90 x - 1)+(\left(- 5 x - 4\right) \left(9 x - 7\right))(-2)=\left(4 - 2 x\right) \left(35 - 45 x\right) + \left(4 - 2 x\right) \left(- 45 x - 36\right) - 2 \left(- 5 x - 4\right) \left(9 x - 7\right) " /> </p> </p>