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Questions: Algebra BusinessCalculus
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Find the derivative of \(\displaystyle y = (2 - 5 x)(4 x - 6)(7 x - 1)\).
Identifying \(\displaystyle f=2 - 5 x\) and \(\displaystyle g=\left(4 x - 6\right) \left(7 x - 1\right)\) and using the product rule with \(\displaystyle f=2 - 5 x \implies f'=-5\). This leaves g as \(\displaystyle g = \left(4 x - 6\right) \left(7 x - 1\right)\) which also requires the product rule. Pushing down in the new product rule \(\displaystyle f=4 x - 6 \implies f'=4\) and \(\displaystyle g=7 x - 1 \implies g'=7\). Popping up a level gives \(\displaystyle g'=(7 x - 1)(4)+(4 x - 6)(7)\)Popping up again (Back to the original problem) gives \(\displaystyle f'=(2 - 5 x)(56 x - 46)+(\left(4 x - 6\right) \left(7 x - 1\right))(-5)=\left(2 - 5 x\right) \left(28 x - 42\right) + \left(2 - 5 x\right) \left(28 x - 4\right) - 5 \left(4 x - 6\right) \left(7 x - 1\right)\)
\begin{question}Find the derivative of $y = (2 - 5 x)(4 x - 6)(7 x - 1)$.
\soln{9cm}{Identifying $f=2 - 5 x$ and $g=\left(4 x - 6\right) \left(7 x - 1\right)$ and using the product rule with $f=2 - 5 x \implies f'=-5$. This leaves g as $g = \left(4 x - 6\right) \left(7 x - 1\right)$ which also requires the product rule. Pushing down in the new product rule $f=4 x - 6 \implies f'=4$ and $g=7 x - 1 \implies g'=7$. Popping up a level gives $g'=(7 x - 1)(4)+(4 x - 6)(7)$Popping up again (Back to the original problem) gives $f'=(2 - 5 x)(56 x - 46)+(\left(4 x - 6\right) \left(7 x - 1\right))(-5)=\left(2 - 5 x\right) \left(28 x - 42\right) + \left(2 - 5 x\right) \left(28 x - 4\right) - 5 \left(4 x - 6\right) \left(7 x - 1\right)$}
\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 = (2 - 5 x)(4 x - 6)(7 x - 1) " src="/equation_images/%20%5Cdisplaystyle%20y%20%3D%20%282%20-%205%20x%29%284%20x%20-%206%29%287%20x%20-%201%29%20" alt="LaTeX: \displaystyle y = (2 - 5 x)(4 x - 6)(7 x - 1) " data-equation-content=" \displaystyle y = (2 - 5 x)(4 x - 6)(7 x - 1) " /> .</p> </p>
<p> <p>Identifying <img class="equation_image" title=" \displaystyle f=2 - 5 x " src="/equation_images/%20%5Cdisplaystyle%20f%3D2%20-%205%20x%20" alt="LaTeX: \displaystyle f=2 - 5 x " data-equation-content=" \displaystyle f=2 - 5 x " /> and <img class="equation_image" title=" \displaystyle g=\left(4 x - 6\right) \left(7 x - 1\right) " src="/equation_images/%20%5Cdisplaystyle%20g%3D%5Cleft%284%20x%20-%206%5Cright%29%20%5Cleft%287%20x%20-%201%5Cright%29%20" alt="LaTeX: \displaystyle g=\left(4 x - 6\right) \left(7 x - 1\right) " data-equation-content=" \displaystyle g=\left(4 x - 6\right) \left(7 x - 1\right) " /> and using the product rule with <img class="equation_image" title=" \displaystyle f=2 - 5 x \implies f'=-5 " src="/equation_images/%20%5Cdisplaystyle%20f%3D2%20-%205%20x%20%5Cimplies%20f%27%3D-5%20" alt="LaTeX: \displaystyle f=2 - 5 x \implies f'=-5 " data-equation-content=" \displaystyle f=2 - 5 x \implies f'=-5 " /> . This leaves g as <img class="equation_image" title=" \displaystyle g = \left(4 x - 6\right) \left(7 x - 1\right) " src="/equation_images/%20%5Cdisplaystyle%20g%20%3D%20%5Cleft%284%20x%20-%206%5Cright%29%20%5Cleft%287%20x%20-%201%5Cright%29%20" alt="LaTeX: \displaystyle g = \left(4 x - 6\right) \left(7 x - 1\right) " data-equation-content=" \displaystyle g = \left(4 x - 6\right) \left(7 x - 1\right) " /> which also requires the product rule. Pushing down in the new product rule <img class="equation_image" title=" \displaystyle f=4 x - 6 \implies f'=4 " src="/equation_images/%20%5Cdisplaystyle%20f%3D4%20x%20-%206%20%5Cimplies%20f%27%3D4%20" alt="LaTeX: \displaystyle f=4 x - 6 \implies f'=4 " data-equation-content=" \displaystyle f=4 x - 6 \implies f'=4 " /> and <img class="equation_image" title=" \displaystyle g=7 x - 1 \implies g'=7 " src="/equation_images/%20%5Cdisplaystyle%20g%3D7%20x%20-%201%20%5Cimplies%20g%27%3D7%20" alt="LaTeX: \displaystyle g=7 x - 1 \implies g'=7 " data-equation-content=" \displaystyle g=7 x - 1 \implies g'=7 " /> . Popping up a level gives <img class="equation_image" title=" \displaystyle g'=(7 x - 1)(4)+(4 x - 6)(7) " src="/equation_images/%20%5Cdisplaystyle%20g%27%3D%287%20x%20-%201%29%284%29%2B%284%20x%20-%206%29%287%29%20" alt="LaTeX: \displaystyle g'=(7 x - 1)(4)+(4 x - 6)(7) " data-equation-content=" \displaystyle g'=(7 x - 1)(4)+(4 x - 6)(7) " /> Popping up again (Back to the original problem) gives <img class="equation_image" title=" \displaystyle f'=(2 - 5 x)(56 x - 46)+(\left(4 x - 6\right) \left(7 x - 1\right))(-5)=\left(2 - 5 x\right) \left(28 x - 42\right) + \left(2 - 5 x\right) \left(28 x - 4\right) - 5 \left(4 x - 6\right) \left(7 x - 1\right) " src="/equation_images/%20%5Cdisplaystyle%20f%27%3D%282%20-%205%20x%29%2856%20x%20-%2046%29%2B%28%5Cleft%284%20x%20-%206%5Cright%29%20%5Cleft%287%20x%20-%201%5Cright%29%29%28-5%29%3D%5Cleft%282%20-%205%20x%5Cright%29%20%5Cleft%2828%20x%20-%2042%5Cright%29%20%2B%20%5Cleft%282%20-%205%20x%5Cright%29%20%5Cleft%2828%20x%20-%204%5Cright%29%20-%205%20%5Cleft%284%20x%20-%206%5Cright%29%20%5Cleft%287%20x%20-%201%5Cright%29%20" alt="LaTeX: \displaystyle f'=(2 - 5 x)(56 x - 46)+(\left(4 x - 6\right) \left(7 x - 1\right))(-5)=\left(2 - 5 x\right) \left(28 x - 42\right) + \left(2 - 5 x\right) \left(28 x - 4\right) - 5 \left(4 x - 6\right) \left(7 x - 1\right) " data-equation-content=" \displaystyle f'=(2 - 5 x)(56 x - 46)+(\left(4 x - 6\right) \left(7 x - 1\right))(-5)=\left(2 - 5 x\right) \left(28 x - 42\right) + \left(2 - 5 x\right) \left(28 x - 4\right) - 5 \left(4 x - 6\right) \left(7 x - 1\right) " /> </p> </p>