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Questions: Algebra BusinessCalculus
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Find the derivative of \(\displaystyle y = (- 4 x^{2} - 8 x - 2)(e^{x})(\sin{\left(x \right)})\).
Identifying \(\displaystyle f=- 4 x^{2} - 8 x - 2\) and \(\displaystyle g=e^{x} \sin{\left(x \right)}\) and using the product rule with \(\displaystyle f=- 4 x^{2} - 8 x - 2 \implies f'=- 8 x - 8\). This leaves g as \(\displaystyle g = e^{x} \sin{\left(x \right)}\) which also requires the product rule. Pushing down in the new product rule \(\displaystyle f=e^{x} \implies f'=e^{x}\) and \(\displaystyle g=\sin{\left(x \right)} \implies g'=\cos{\left(x \right)}\). Popping up a level gives \(\displaystyle g'=(\sin{\left(x \right)})(e^{x})+(e^{x})(\cos{\left(x \right)})\)Popping up again (Back to the original problem) gives \(\displaystyle f'=(- 4 x^{2} - 8 x - 2)(e^{x} \sin{\left(x \right)} + e^{x} \cos{\left(x \right)})+(e^{x} \sin{\left(x \right)})(- 8 x - 8)=\left(- 8 x - 8\right) e^{x} \sin{\left(x \right)} + \left(- 4 x^{2} - 8 x - 2\right) e^{x} \sin{\left(x \right)} + \left(- 4 x^{2} - 8 x - 2\right) e^{x} \cos{\left(x \right)}\)
\begin{question}Find the derivative of $y = (- 4 x^{2} - 8 x - 2)(e^{x})(\sin{\left(x \right)})$.
\soln{9cm}{Identifying $f=- 4 x^{2} - 8 x - 2$ and $g=e^{x} \sin{\left(x \right)}$ and using the product rule with $f=- 4 x^{2} - 8 x - 2 \implies f'=- 8 x - 8$. This leaves g as $g = e^{x} \sin{\left(x \right)}$ which also requires the product rule. Pushing down in the new product rule $f=e^{x} \implies f'=e^{x}$ and $g=\sin{\left(x \right)} \implies g'=\cos{\left(x \right)}$. Popping up a level gives $g'=(\sin{\left(x \right)})(e^{x})+(e^{x})(\cos{\left(x \right)})$Popping up again (Back to the original problem) gives $f'=(- 4 x^{2} - 8 x - 2)(e^{x} \sin{\left(x \right)} + e^{x} \cos{\left(x \right)})+(e^{x} \sin{\left(x \right)})(- 8 x - 8)=\left(- 8 x - 8\right) e^{x} \sin{\left(x \right)} + \left(- 4 x^{2} - 8 x - 2\right) e^{x} \sin{\left(x \right)} + \left(- 4 x^{2} - 8 x - 2\right) e^{x} \cos{\left(x \right)}$}
\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 = (- 4 x^{2} - 8 x - 2)(e^{x})(\sin{\left(x \right)}) " src="/equation_images/%20%5Cdisplaystyle%20y%20%3D%20%28-%204%20x%5E%7B2%7D%20-%208%20x%20-%202%29%28e%5E%7Bx%7D%29%28%5Csin%7B%5Cleft%28x%20%5Cright%29%7D%29%20" alt="LaTeX: \displaystyle y = (- 4 x^{2} - 8 x - 2)(e^{x})(\sin{\left(x \right)}) " data-equation-content=" \displaystyle y = (- 4 x^{2} - 8 x - 2)(e^{x})(\sin{\left(x \right)}) " /> .</p> </p><p> <p>Identifying <img class="equation_image" title=" \displaystyle f=- 4 x^{2} - 8 x - 2 " src="/equation_images/%20%5Cdisplaystyle%20f%3D-%204%20x%5E%7B2%7D%20-%208%20x%20-%202%20" alt="LaTeX: \displaystyle f=- 4 x^{2} - 8 x - 2 " data-equation-content=" \displaystyle f=- 4 x^{2} - 8 x - 2 " /> and <img class="equation_image" title=" \displaystyle g=e^{x} \sin{\left(x \right)} " src="/equation_images/%20%5Cdisplaystyle%20g%3De%5E%7Bx%7D%20%5Csin%7B%5Cleft%28x%20%5Cright%29%7D%20" alt="LaTeX: \displaystyle g=e^{x} \sin{\left(x \right)} " data-equation-content=" \displaystyle g=e^{x} \sin{\left(x \right)} " /> and using the product rule with <img class="equation_image" title=" \displaystyle f=- 4 x^{2} - 8 x - 2 \implies f'=- 8 x - 8 " src="/equation_images/%20%5Cdisplaystyle%20f%3D-%204%20x%5E%7B2%7D%20-%208%20x%20-%202%20%5Cimplies%20f%27%3D-%208%20x%20-%208%20" alt="LaTeX: \displaystyle f=- 4 x^{2} - 8 x - 2 \implies f'=- 8 x - 8 " data-equation-content=" \displaystyle f=- 4 x^{2} - 8 x - 2 \implies f'=- 8 x - 8 " /> . This leaves g as <img class="equation_image" title=" \displaystyle g = e^{x} \sin{\left(x \right)} " src="/equation_images/%20%5Cdisplaystyle%20g%20%3D%20e%5E%7Bx%7D%20%5Csin%7B%5Cleft%28x%20%5Cright%29%7D%20" alt="LaTeX: \displaystyle g = e^{x} \sin{\left(x \right)} " data-equation-content=" \displaystyle g = e^{x} \sin{\left(x \right)} " /> which also requires the product rule. Pushing down in the new product rule <img class="equation_image" title=" \displaystyle f=e^{x} \implies f'=e^{x} " src="/equation_images/%20%5Cdisplaystyle%20f%3De%5E%7Bx%7D%20%5Cimplies%20f%27%3De%5E%7Bx%7D%20" alt="LaTeX: \displaystyle f=e^{x} \implies f'=e^{x} " data-equation-content=" \displaystyle f=e^{x} \implies f'=e^{x} " /> and <img class="equation_image" title=" \displaystyle g=\sin{\left(x \right)} \implies g'=\cos{\left(x \right)} " src="/equation_images/%20%5Cdisplaystyle%20g%3D%5Csin%7B%5Cleft%28x%20%5Cright%29%7D%20%5Cimplies%20g%27%3D%5Ccos%7B%5Cleft%28x%20%5Cright%29%7D%20" alt="LaTeX: \displaystyle g=\sin{\left(x \right)} \implies g'=\cos{\left(x \right)} " data-equation-content=" \displaystyle g=\sin{\left(x \right)} \implies g'=\cos{\left(x \right)} " /> . Popping up a level gives <img class="equation_image" title=" \displaystyle g'=(\sin{\left(x \right)})(e^{x})+(e^{x})(\cos{\left(x \right)}) " src="/equation_images/%20%5Cdisplaystyle%20g%27%3D%28%5Csin%7B%5Cleft%28x%20%5Cright%29%7D%29%28e%5E%7Bx%7D%29%2B%28e%5E%7Bx%7D%29%28%5Ccos%7B%5Cleft%28x%20%5Cright%29%7D%29%20" alt="LaTeX: \displaystyle g'=(\sin{\left(x \right)})(e^{x})+(e^{x})(\cos{\left(x \right)}) " data-equation-content=" \displaystyle g'=(\sin{\left(x \right)})(e^{x})+(e^{x})(\cos{\left(x \right)}) " /> Popping up again (Back to the original problem) gives <img class="equation_image" title=" \displaystyle f'=(- 4 x^{2} - 8 x - 2)(e^{x} \sin{\left(x \right)} + e^{x} \cos{\left(x \right)})+(e^{x} \sin{\left(x \right)})(- 8 x - 8)=\left(- 8 x - 8\right) e^{x} \sin{\left(x \right)} + \left(- 4 x^{2} - 8 x - 2\right) e^{x} \sin{\left(x \right)} + \left(- 4 x^{2} - 8 x - 2\right) e^{x} \cos{\left(x \right)} " src="/equation_images/%20%5Cdisplaystyle%20f%27%3D%28-%204%20x%5E%7B2%7D%20-%208%20x%20-%202%29%28e%5E%7Bx%7D%20%5Csin%7B%5Cleft%28x%20%5Cright%29%7D%20%2B%20e%5E%7Bx%7D%20%5Ccos%7B%5Cleft%28x%20%5Cright%29%7D%29%2B%28e%5E%7Bx%7D%20%5Csin%7B%5Cleft%28x%20%5Cright%29%7D%29%28-%208%20x%20-%208%29%3D%5Cleft%28-%208%20x%20-%208%5Cright%29%20e%5E%7Bx%7D%20%5Csin%7B%5Cleft%28x%20%5Cright%29%7D%20%2B%20%5Cleft%28-%204%20x%5E%7B2%7D%20-%208%20x%20-%202%5Cright%29%20e%5E%7Bx%7D%20%5Csin%7B%5Cleft%28x%20%5Cright%29%7D%20%2B%20%5Cleft%28-%204%20x%5E%7B2%7D%20-%208%20x%20-%202%5Cright%29%20e%5E%7Bx%7D%20%5Ccos%7B%5Cleft%28x%20%5Cright%29%7D%20" alt="LaTeX: \displaystyle f'=(- 4 x^{2} - 8 x - 2)(e^{x} \sin{\left(x \right)} + e^{x} \cos{\left(x \right)})+(e^{x} \sin{\left(x \right)})(- 8 x - 8)=\left(- 8 x - 8\right) e^{x} \sin{\left(x \right)} + \left(- 4 x^{2} - 8 x - 2\right) e^{x} \sin{\left(x \right)} + \left(- 4 x^{2} - 8 x - 2\right) e^{x} \cos{\left(x \right)} " data-equation-content=" \displaystyle f'=(- 4 x^{2} - 8 x - 2)(e^{x} \sin{\left(x \right)} + e^{x} \cos{\left(x \right)})+(e^{x} \sin{\left(x \right)})(- 8 x - 8)=\left(- 8 x - 8\right) e^{x} \sin{\left(x \right)} + \left(- 4 x^{2} - 8 x - 2\right) e^{x} \sin{\left(x \right)} + \left(- 4 x^{2} - 8 x - 2\right) e^{x} \cos{\left(x \right)} " /> </p> </p>