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Questions: Algebra BusinessCalculus
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Find the derivative of \(\displaystyle y = (e^{x})(\sin{\left(x \right)})(8 x^{2} + 6 x - 8)\).
Identifying \(\displaystyle f=e^{x}\) and \(\displaystyle g=\left(8 x^{2} + 6 x - 8\right) \sin{\left(x \right)}\) and using the product rule with \(\displaystyle f=e^{x} \implies f'=e^{x}\). This leaves g as \(\displaystyle g = \left(8 x^{2} + 6 x - 8\right) \sin{\left(x \right)}\) which also requires the product rule. Pushing down in the new product rule \(\displaystyle f=\sin{\left(x \right)} \implies f'=\cos{\left(x \right)}\) and \(\displaystyle g=8 x^{2} + 6 x - 8 \implies g'=16 x + 6\). Popping up a level gives \(\displaystyle g'=(8 x^{2} + 6 x - 8)(\cos{\left(x \right)})+(\sin{\left(x \right)})(16 x + 6)\)Popping up again (Back to the original problem) gives \(\displaystyle f'=(e^{x})(\left(16 x + 6\right) \sin{\left(x \right)} + \left(8 x^{2} + 6 x - 8\right) \cos{\left(x \right)})+(\left(8 x^{2} + 6 x - 8\right) \sin{\left(x \right)})(e^{x})=\left(16 x + 6\right) e^{x} \sin{\left(x \right)} + \left(8 x^{2} + 6 x - 8\right) e^{x} \sin{\left(x \right)} + \left(8 x^{2} + 6 x - 8\right) e^{x} \cos{\left(x \right)}\)
\begin{question}Find the derivative of $y = (e^{x})(\sin{\left(x \right)})(8 x^{2} + 6 x - 8)$.
\soln{9cm}{Identifying $f=e^{x}$ and $g=\left(8 x^{2} + 6 x - 8\right) \sin{\left(x \right)}$ and using the product rule with $f=e^{x} \implies f'=e^{x}$. This leaves g as $g = \left(8 x^{2} + 6 x - 8\right) \sin{\left(x \right)}$ which also requires the product rule. Pushing down in the new product rule $f=\sin{\left(x \right)} \implies f'=\cos{\left(x \right)}$ and $g=8 x^{2} + 6 x - 8 \implies g'=16 x + 6$. Popping up a level gives $g'=(8 x^{2} + 6 x - 8)(\cos{\left(x \right)})+(\sin{\left(x \right)})(16 x + 6)$Popping up again (Back to the original problem) gives $f'=(e^{x})(\left(16 x + 6\right) \sin{\left(x \right)} + \left(8 x^{2} + 6 x - 8\right) \cos{\left(x \right)})+(\left(8 x^{2} + 6 x - 8\right) \sin{\left(x \right)})(e^{x})=\left(16 x + 6\right) e^{x} \sin{\left(x \right)} + \left(8 x^{2} + 6 x - 8\right) e^{x} \sin{\left(x \right)} + \left(8 x^{2} + 6 x - 8\right) e^{x} \cos{\left(x \right)}$}
\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 = (e^{x})(\sin{\left(x \right)})(8 x^{2} + 6 x - 8) " src="/equation_images/%20%5Cdisplaystyle%20y%20%3D%20%28e%5E%7Bx%7D%29%28%5Csin%7B%5Cleft%28x%20%5Cright%29%7D%29%288%20x%5E%7B2%7D%20%2B%206%20x%20-%208%29%20" alt="LaTeX: \displaystyle y = (e^{x})(\sin{\left(x \right)})(8 x^{2} + 6 x - 8) " data-equation-content=" \displaystyle y = (e^{x})(\sin{\left(x \right)})(8 x^{2} + 6 x - 8) " /> .</p> </p><p> <p>Identifying <img class="equation_image" title=" \displaystyle f=e^{x} " src="/equation_images/%20%5Cdisplaystyle%20f%3De%5E%7Bx%7D%20" alt="LaTeX: \displaystyle f=e^{x} " data-equation-content=" \displaystyle f=e^{x} " /> and <img class="equation_image" title=" \displaystyle g=\left(8 x^{2} + 6 x - 8\right) \sin{\left(x \right)} " src="/equation_images/%20%5Cdisplaystyle%20g%3D%5Cleft%288%20x%5E%7B2%7D%20%2B%206%20x%20-%208%5Cright%29%20%5Csin%7B%5Cleft%28x%20%5Cright%29%7D%20" alt="LaTeX: \displaystyle g=\left(8 x^{2} + 6 x - 8\right) \sin{\left(x \right)} " data-equation-content=" \displaystyle g=\left(8 x^{2} + 6 x - 8\right) \sin{\left(x \right)} " /> and using the product rule with <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} " /> . This leaves g as <img class="equation_image" title=" \displaystyle g = \left(8 x^{2} + 6 x - 8\right) \sin{\left(x \right)} " src="/equation_images/%20%5Cdisplaystyle%20g%20%3D%20%5Cleft%288%20x%5E%7B2%7D%20%2B%206%20x%20-%208%5Cright%29%20%5Csin%7B%5Cleft%28x%20%5Cright%29%7D%20" alt="LaTeX: \displaystyle g = \left(8 x^{2} + 6 x - 8\right) \sin{\left(x \right)} " data-equation-content=" \displaystyle g = \left(8 x^{2} + 6 x - 8\right) \sin{\left(x \right)} " /> which also requires the product rule. Pushing down in the new product rule <img class="equation_image" title=" \displaystyle f=\sin{\left(x \right)} \implies f'=\cos{\left(x \right)} " src="/equation_images/%20%5Cdisplaystyle%20f%3D%5Csin%7B%5Cleft%28x%20%5Cright%29%7D%20%5Cimplies%20f%27%3D%5Ccos%7B%5Cleft%28x%20%5Cright%29%7D%20" alt="LaTeX: \displaystyle f=\sin{\left(x \right)} \implies f'=\cos{\left(x \right)} " data-equation-content=" \displaystyle f=\sin{\left(x \right)} \implies f'=\cos{\left(x \right)} " /> and <img class="equation_image" title=" \displaystyle g=8 x^{2} + 6 x - 8 \implies g'=16 x + 6 " src="/equation_images/%20%5Cdisplaystyle%20g%3D8%20x%5E%7B2%7D%20%2B%206%20x%20-%208%20%5Cimplies%20g%27%3D16%20x%20%2B%206%20" alt="LaTeX: \displaystyle g=8 x^{2} + 6 x - 8 \implies g'=16 x + 6 " data-equation-content=" \displaystyle g=8 x^{2} + 6 x - 8 \implies g'=16 x + 6 " /> . Popping up a level gives <img class="equation_image" title=" \displaystyle g'=(8 x^{2} + 6 x - 8)(\cos{\left(x \right)})+(\sin{\left(x \right)})(16 x + 6) " src="/equation_images/%20%5Cdisplaystyle%20g%27%3D%288%20x%5E%7B2%7D%20%2B%206%20x%20-%208%29%28%5Ccos%7B%5Cleft%28x%20%5Cright%29%7D%29%2B%28%5Csin%7B%5Cleft%28x%20%5Cright%29%7D%29%2816%20x%20%2B%206%29%20" alt="LaTeX: \displaystyle g'=(8 x^{2} + 6 x - 8)(\cos{\left(x \right)})+(\sin{\left(x \right)})(16 x + 6) " data-equation-content=" \displaystyle g'=(8 x^{2} + 6 x - 8)(\cos{\left(x \right)})+(\sin{\left(x \right)})(16 x + 6) " /> Popping up again (Back to the original problem) gives <img class="equation_image" title=" \displaystyle f'=(e^{x})(\left(16 x + 6\right) \sin{\left(x \right)} + \left(8 x^{2} + 6 x - 8\right) \cos{\left(x \right)})+(\left(8 x^{2} + 6 x - 8\right) \sin{\left(x \right)})(e^{x})=\left(16 x + 6\right) e^{x} \sin{\left(x \right)} + \left(8 x^{2} + 6 x - 8\right) e^{x} \sin{\left(x \right)} + \left(8 x^{2} + 6 x - 8\right) e^{x} \cos{\left(x \right)} " src="/equation_images/%20%5Cdisplaystyle%20f%27%3D%28e%5E%7Bx%7D%29%28%5Cleft%2816%20x%20%2B%206%5Cright%29%20%5Csin%7B%5Cleft%28x%20%5Cright%29%7D%20%2B%20%5Cleft%288%20x%5E%7B2%7D%20%2B%206%20x%20-%208%5Cright%29%20%5Ccos%7B%5Cleft%28x%20%5Cright%29%7D%29%2B%28%5Cleft%288%20x%5E%7B2%7D%20%2B%206%20x%20-%208%5Cright%29%20%5Csin%7B%5Cleft%28x%20%5Cright%29%7D%29%28e%5E%7Bx%7D%29%3D%5Cleft%2816%20x%20%2B%206%5Cright%29%20e%5E%7Bx%7D%20%5Csin%7B%5Cleft%28x%20%5Cright%29%7D%20%2B%20%5Cleft%288%20x%5E%7B2%7D%20%2B%206%20x%20-%208%5Cright%29%20e%5E%7Bx%7D%20%5Csin%7B%5Cleft%28x%20%5Cright%29%7D%20%2B%20%5Cleft%288%20x%5E%7B2%7D%20%2B%206%20x%20-%208%5Cright%29%20e%5E%7Bx%7D%20%5Ccos%7B%5Cleft%28x%20%5Cright%29%7D%20" alt="LaTeX: \displaystyle f'=(e^{x})(\left(16 x + 6\right) \sin{\left(x \right)} + \left(8 x^{2} + 6 x - 8\right) \cos{\left(x \right)})+(\left(8 x^{2} + 6 x - 8\right) \sin{\left(x \right)})(e^{x})=\left(16 x + 6\right) e^{x} \sin{\left(x \right)} + \left(8 x^{2} + 6 x - 8\right) e^{x} \sin{\left(x \right)} + \left(8 x^{2} + 6 x - 8\right) e^{x} \cos{\left(x \right)} " data-equation-content=" \displaystyle f'=(e^{x})(\left(16 x + 6\right) \sin{\left(x \right)} + \left(8 x^{2} + 6 x - 8\right) \cos{\left(x \right)})+(\left(8 x^{2} + 6 x - 8\right) \sin{\left(x \right)})(e^{x})=\left(16 x + 6\right) e^{x} \sin{\left(x \right)} + \left(8 x^{2} + 6 x - 8\right) e^{x} \sin{\left(x \right)} + \left(8 x^{2} + 6 x - 8\right) e^{x} \cos{\left(x \right)} " /> </p> </p>