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Questions: Algebra BusinessCalculus
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Find the derivative of \(\displaystyle y = (5 x^{2} - 9 x + 4)(5 x^{2} - 7 x - 2)(\cos{\left(x \right)})\).
Identifying \(\displaystyle f=5 x^{2} - 9 x + 4\) and \(\displaystyle g=\left(5 x^{2} - 7 x - 2\right) \cos{\left(x \right)}\) and using the product rule with \(\displaystyle f=5 x^{2} - 9 x + 4 \implies f'=10 x - 9\). This leaves g as \(\displaystyle g = \left(5 x^{2} - 7 x - 2\right) \cos{\left(x \right)}\) which also requires the product rule. Pushing down in the new product rule \(\displaystyle f=5 x^{2} - 7 x - 2 \implies f'=10 x - 7\) and \(\displaystyle g=\cos{\left(x \right)} \implies g'=- \sin{\left(x \right)}\). Popping up a level gives \(\displaystyle g'=(\cos{\left(x \right)})(10 x - 7)+(5 x^{2} - 7 x - 2)(- \sin{\left(x \right)})\)Popping up again (Back to the original problem) gives \(\displaystyle f'=(5 x^{2} - 9 x + 4)(\left(10 x - 7\right) \cos{\left(x \right)} - \left(5 x^{2} - 7 x - 2\right) \sin{\left(x \right)})+(\left(5 x^{2} - 7 x - 2\right) \cos{\left(x \right)})(10 x - 9)=\left(10 x - 9\right) \left(5 x^{2} - 7 x - 2\right) \cos{\left(x \right)} + \left(10 x - 7\right) \left(5 x^{2} - 9 x + 4\right) \cos{\left(x \right)} - \left(5 x^{2} - 9 x + 4\right) \left(5 x^{2} - 7 x - 2\right) \sin{\left(x \right)}\)
\begin{question}Find the derivative of $y = (5 x^{2} - 9 x + 4)(5 x^{2} - 7 x - 2)(\cos{\left(x \right)})$.
\soln{9cm}{Identifying $f=5 x^{2} - 9 x + 4$ and $g=\left(5 x^{2} - 7 x - 2\right) \cos{\left(x \right)}$ and using the product rule with $f=5 x^{2} - 9 x + 4 \implies f'=10 x - 9$. This leaves g as $g = \left(5 x^{2} - 7 x - 2\right) \cos{\left(x \right)}$ which also requires the product rule. Pushing down in the new product rule $f=5 x^{2} - 7 x - 2 \implies f'=10 x - 7$ and $g=\cos{\left(x \right)} \implies g'=- \sin{\left(x \right)}$. Popping up a level gives $g'=(\cos{\left(x \right)})(10 x - 7)+(5 x^{2} - 7 x - 2)(- \sin{\left(x \right)})$Popping up again (Back to the original problem) gives $f'=(5 x^{2} - 9 x + 4)(\left(10 x - 7\right) \cos{\left(x \right)} - \left(5 x^{2} - 7 x - 2\right) \sin{\left(x \right)})+(\left(5 x^{2} - 7 x - 2\right) \cos{\left(x \right)})(10 x - 9)=\left(10 x - 9\right) \left(5 x^{2} - 7 x - 2\right) \cos{\left(x \right)} + \left(10 x - 7\right) \left(5 x^{2} - 9 x + 4\right) \cos{\left(x \right)} - \left(5 x^{2} - 9 x + 4\right) \left(5 x^{2} - 7 x - 2\right) \sin{\left(x \right)}$}
\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 = (5 x^{2} - 9 x + 4)(5 x^{2} - 7 x - 2)(\cos{\left(x \right)}) " src="/equation_images/%20%5Cdisplaystyle%20y%20%3D%20%285%20x%5E%7B2%7D%20-%209%20x%20%2B%204%29%285%20x%5E%7B2%7D%20-%207%20x%20-%202%29%28%5Ccos%7B%5Cleft%28x%20%5Cright%29%7D%29%20" alt="LaTeX: \displaystyle y = (5 x^{2} - 9 x + 4)(5 x^{2} - 7 x - 2)(\cos{\left(x \right)}) " data-equation-content=" \displaystyle y = (5 x^{2} - 9 x + 4)(5 x^{2} - 7 x - 2)(\cos{\left(x \right)}) " /> .</p> </p><p> <p>Identifying <img class="equation_image" title=" \displaystyle f=5 x^{2} - 9 x + 4 " src="/equation_images/%20%5Cdisplaystyle%20f%3D5%20x%5E%7B2%7D%20-%209%20x%20%2B%204%20" alt="LaTeX: \displaystyle f=5 x^{2} - 9 x + 4 " data-equation-content=" \displaystyle f=5 x^{2} - 9 x + 4 " /> and <img class="equation_image" title=" \displaystyle g=\left(5 x^{2} - 7 x - 2\right) \cos{\left(x \right)} " src="/equation_images/%20%5Cdisplaystyle%20g%3D%5Cleft%285%20x%5E%7B2%7D%20-%207%20x%20-%202%5Cright%29%20%5Ccos%7B%5Cleft%28x%20%5Cright%29%7D%20" alt="LaTeX: \displaystyle g=\left(5 x^{2} - 7 x - 2\right) \cos{\left(x \right)} " data-equation-content=" \displaystyle g=\left(5 x^{2} - 7 x - 2\right) \cos{\left(x \right)} " /> and using the product rule with <img class="equation_image" title=" \displaystyle f=5 x^{2} - 9 x + 4 \implies f'=10 x - 9 " src="/equation_images/%20%5Cdisplaystyle%20f%3D5%20x%5E%7B2%7D%20-%209%20x%20%2B%204%20%5Cimplies%20f%27%3D10%20x%20-%209%20" alt="LaTeX: \displaystyle f=5 x^{2} - 9 x + 4 \implies f'=10 x - 9 " data-equation-content=" \displaystyle f=5 x^{2} - 9 x + 4 \implies f'=10 x - 9 " /> . This leaves g as <img class="equation_image" title=" \displaystyle g = \left(5 x^{2} - 7 x - 2\right) \cos{\left(x \right)} " src="/equation_images/%20%5Cdisplaystyle%20g%20%3D%20%5Cleft%285%20x%5E%7B2%7D%20-%207%20x%20-%202%5Cright%29%20%5Ccos%7B%5Cleft%28x%20%5Cright%29%7D%20" alt="LaTeX: \displaystyle g = \left(5 x^{2} - 7 x - 2\right) \cos{\left(x \right)} " data-equation-content=" \displaystyle g = \left(5 x^{2} - 7 x - 2\right) \cos{\left(x \right)} " /> which also requires the product rule. Pushing down in the new product rule <img class="equation_image" title=" \displaystyle f=5 x^{2} - 7 x - 2 \implies f'=10 x - 7 " src="/equation_images/%20%5Cdisplaystyle%20f%3D5%20x%5E%7B2%7D%20-%207%20x%20-%202%20%5Cimplies%20f%27%3D10%20x%20-%207%20" alt="LaTeX: \displaystyle f=5 x^{2} - 7 x - 2 \implies f'=10 x - 7 " data-equation-content=" \displaystyle f=5 x^{2} - 7 x - 2 \implies f'=10 x - 7 " /> and <img class="equation_image" title=" \displaystyle g=\cos{\left(x \right)} \implies g'=- \sin{\left(x \right)} " src="/equation_images/%20%5Cdisplaystyle%20g%3D%5Ccos%7B%5Cleft%28x%20%5Cright%29%7D%20%5Cimplies%20g%27%3D-%20%5Csin%7B%5Cleft%28x%20%5Cright%29%7D%20" alt="LaTeX: \displaystyle g=\cos{\left(x \right)} \implies g'=- \sin{\left(x \right)} " data-equation-content=" \displaystyle g=\cos{\left(x \right)} \implies g'=- \sin{\left(x \right)} " /> . Popping up a level gives <img class="equation_image" title=" \displaystyle g'=(\cos{\left(x \right)})(10 x - 7)+(5 x^{2} - 7 x - 2)(- \sin{\left(x \right)}) " src="/equation_images/%20%5Cdisplaystyle%20g%27%3D%28%5Ccos%7B%5Cleft%28x%20%5Cright%29%7D%29%2810%20x%20-%207%29%2B%285%20x%5E%7B2%7D%20-%207%20x%20-%202%29%28-%20%5Csin%7B%5Cleft%28x%20%5Cright%29%7D%29%20" alt="LaTeX: \displaystyle g'=(\cos{\left(x \right)})(10 x - 7)+(5 x^{2} - 7 x - 2)(- \sin{\left(x \right)}) " data-equation-content=" \displaystyle g'=(\cos{\left(x \right)})(10 x - 7)+(5 x^{2} - 7 x - 2)(- \sin{\left(x \right)}) " /> Popping up again (Back to the original problem) gives <img class="equation_image" title=" \displaystyle f'=(5 x^{2} - 9 x + 4)(\left(10 x - 7\right) \cos{\left(x \right)} - \left(5 x^{2} - 7 x - 2\right) \sin{\left(x \right)})+(\left(5 x^{2} - 7 x - 2\right) \cos{\left(x \right)})(10 x - 9)=\left(10 x - 9\right) \left(5 x^{2} - 7 x - 2\right) \cos{\left(x \right)} + \left(10 x - 7\right) \left(5 x^{2} - 9 x + 4\right) \cos{\left(x \right)} - \left(5 x^{2} - 9 x + 4\right) \left(5 x^{2} - 7 x - 2\right) \sin{\left(x \right)} " src="/equation_images/%20%5Cdisplaystyle%20f%27%3D%285%20x%5E%7B2%7D%20-%209%20x%20%2B%204%29%28%5Cleft%2810%20x%20-%207%5Cright%29%20%5Ccos%7B%5Cleft%28x%20%5Cright%29%7D%20-%20%5Cleft%285%20x%5E%7B2%7D%20-%207%20x%20-%202%5Cright%29%20%5Csin%7B%5Cleft%28x%20%5Cright%29%7D%29%2B%28%5Cleft%285%20x%5E%7B2%7D%20-%207%20x%20-%202%5Cright%29%20%5Ccos%7B%5Cleft%28x%20%5Cright%29%7D%29%2810%20x%20-%209%29%3D%5Cleft%2810%20x%20-%209%5Cright%29%20%5Cleft%285%20x%5E%7B2%7D%20-%207%20x%20-%202%5Cright%29%20%5Ccos%7B%5Cleft%28x%20%5Cright%29%7D%20%2B%20%5Cleft%2810%20x%20-%207%5Cright%29%20%5Cleft%285%20x%5E%7B2%7D%20-%209%20x%20%2B%204%5Cright%29%20%5Ccos%7B%5Cleft%28x%20%5Cright%29%7D%20-%20%5Cleft%285%20x%5E%7B2%7D%20-%209%20x%20%2B%204%5Cright%29%20%5Cleft%285%20x%5E%7B2%7D%20-%207%20x%20-%202%5Cright%29%20%5Csin%7B%5Cleft%28x%20%5Cright%29%7D%20" alt="LaTeX: \displaystyle f'=(5 x^{2} - 9 x + 4)(\left(10 x - 7\right) \cos{\left(x \right)} - \left(5 x^{2} - 7 x - 2\right) \sin{\left(x \right)})+(\left(5 x^{2} - 7 x - 2\right) \cos{\left(x \right)})(10 x - 9)=\left(10 x - 9\right) \left(5 x^{2} - 7 x - 2\right) \cos{\left(x \right)} + \left(10 x - 7\right) \left(5 x^{2} - 9 x + 4\right) \cos{\left(x \right)} - \left(5 x^{2} - 9 x + 4\right) \left(5 x^{2} - 7 x - 2\right) \sin{\left(x \right)} " data-equation-content=" \displaystyle f'=(5 x^{2} - 9 x + 4)(\left(10 x - 7\right) \cos{\left(x \right)} - \left(5 x^{2} - 7 x - 2\right) \sin{\left(x \right)})+(\left(5 x^{2} - 7 x - 2\right) \cos{\left(x \right)})(10 x - 9)=\left(10 x - 9\right) \left(5 x^{2} - 7 x - 2\right) \cos{\left(x \right)} + \left(10 x - 7\right) \left(5 x^{2} - 9 x + 4\right) \cos{\left(x \right)} - \left(5 x^{2} - 9 x + 4\right) \left(5 x^{2} - 7 x - 2\right) \sin{\left(x \right)} " /> </p> </p>