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Identifying \(\displaystyle f=7 - 5 x\) and \(\displaystyle g=\log{\left(x \right)} \sin{\left(x \right)}\) and using the product rule with \(\displaystyle f=7 - 5 x \implies f'=-5\). This leaves g as \(\displaystyle g = \log{\left(x \right)} \sin{\left(x \right)}\) which also requires the product rule. Pushing down in the new product rule \(\displaystyle f=\log{\left(x \right)} \implies f'=\frac{1}{x}\) and \(\displaystyle g=\sin{\left(x \right)} \implies g'=\cos{\left(x \right)}\). Popping up a level gives \(\displaystyle g'=(\sin{\left(x \right)})(\frac{1}{x})+(\log{\left(x \right)})(\cos{\left(x \right)})\)Popping up again (Back to the original problem) gives \(\displaystyle f'=(7 - 5 x)(\log{\left(x \right)} \cos{\left(x \right)} + \frac{\sin{\left(x \right)}}{x})+(\log{\left(x \right)} \sin{\left(x \right)})(-5)=\left(7 - 5 x\right) \log{\left(x \right)} \cos{\left(x \right)} - 5 \log{\left(x \right)} \sin{\left(x \right)} + \frac{\left(7 - 5 x\right) \sin{\left(x \right)}}{x}\)

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\begin{question}Find the derivative of $y = (7 - 5 x)(\log{\left(x \right)})(\sin{\left(x \right)})$.
    \soln{9cm}{Identifying $f=7 - 5 x$ and $g=\log{\left(x \right)} \sin{\left(x \right)}$ and using the product rule with $f=7 - 5 x \implies f'=-5$. This leaves g as $g = \log{\left(x \right)} \sin{\left(x \right)}$ which also requires the product rule. Pushing down in the new product rule $f=\log{\left(x \right)} \implies f'=\frac{1}{x}$ and $g=\sin{\left(x \right)} \implies g'=\cos{\left(x \right)}$. Popping up a level gives $g'=(\sin{\left(x \right)})(\frac{1}{x})+(\log{\left(x \right)})(\cos{\left(x \right)})$Popping up again (Back to the original problem) gives $f'=(7 - 5 x)(\log{\left(x \right)} \cos{\left(x \right)} + \frac{\sin{\left(x \right)}}{x})+(\log{\left(x \right)} \sin{\left(x \right)})(-5)=\left(7 - 5 x\right) \log{\left(x \right)} \cos{\left(x \right)} - 5 \log{\left(x \right)} \sin{\left(x \right)} + \frac{\left(7 - 5 x\right) \sin{\left(x \right)}}{x}$}

\end{question}

Download Question and Solution Environment\(\LaTeX\)

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\begin{document}\begin{question}(10pts) The question goes here!
    \soln{9cm}{The solution goes here.}

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HTML for Canvas

<p> <p>Find the derivative of  <img class="equation_image" title=" \displaystyle y = (7 - 5 x)(\log{\left(x \right)})(\sin{\left(x \right)}) " src="/equation_images/%20%5Cdisplaystyle%20y%20%3D%20%287%20-%205%20x%29%28%5Clog%7B%5Cleft%28x%20%5Cright%29%7D%29%28%5Csin%7B%5Cleft%28x%20%5Cright%29%7D%29%20" alt="LaTeX:  \displaystyle y = (7 - 5 x)(\log{\left(x \right)})(\sin{\left(x \right)}) " data-equation-content=" \displaystyle y = (7 - 5 x)(\log{\left(x \right)})(\sin{\left(x \right)}) " /> .</p> </p>

HTML for Canvas

<p> <p>Identifying  <img class="equation_image" title=" \displaystyle f=7 - 5 x " src="/equation_images/%20%5Cdisplaystyle%20f%3D7%20-%205%20x%20" alt="LaTeX:  \displaystyle f=7 - 5 x " data-equation-content=" \displaystyle f=7 - 5 x " />  and  <img class="equation_image" title=" \displaystyle g=\log{\left(x \right)} \sin{\left(x \right)} " src="/equation_images/%20%5Cdisplaystyle%20g%3D%5Clog%7B%5Cleft%28x%20%5Cright%29%7D%20%5Csin%7B%5Cleft%28x%20%5Cright%29%7D%20" alt="LaTeX:  \displaystyle g=\log{\left(x \right)} \sin{\left(x \right)} " data-equation-content=" \displaystyle g=\log{\left(x \right)} \sin{\left(x \right)} " />  and using the product rule with  <img class="equation_image" title=" \displaystyle f=7 - 5 x \implies f'=-5 " src="/equation_images/%20%5Cdisplaystyle%20f%3D7%20-%205%20x%20%5Cimplies%20f%27%3D-5%20" alt="LaTeX:  \displaystyle f=7 - 5 x \implies f'=-5 " data-equation-content=" \displaystyle f=7 - 5 x \implies f'=-5 " /> . This leaves g as  <img class="equation_image" title=" \displaystyle g = \log{\left(x \right)} \sin{\left(x \right)} " src="/equation_images/%20%5Cdisplaystyle%20g%20%3D%20%5Clog%7B%5Cleft%28x%20%5Cright%29%7D%20%5Csin%7B%5Cleft%28x%20%5Cright%29%7D%20" alt="LaTeX:  \displaystyle g = \log{\left(x \right)} \sin{\left(x \right)} " data-equation-content=" \displaystyle g = \log{\left(x \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=\log{\left(x \right)} \implies f'=\frac{1}{x} " src="/equation_images/%20%5Cdisplaystyle%20f%3D%5Clog%7B%5Cleft%28x%20%5Cright%29%7D%20%5Cimplies%20f%27%3D%5Cfrac%7B1%7D%7Bx%7D%20" alt="LaTeX:  \displaystyle f=\log{\left(x \right)} \implies f'=\frac{1}{x} " data-equation-content=" \displaystyle f=\log{\left(x \right)} \implies f'=\frac{1}{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)})(\frac{1}{x})+(\log{\left(x \right)})(\cos{\left(x \right)}) " src="/equation_images/%20%5Cdisplaystyle%20g%27%3D%28%5Csin%7B%5Cleft%28x%20%5Cright%29%7D%29%28%5Cfrac%7B1%7D%7Bx%7D%29%2B%28%5Clog%7B%5Cleft%28x%20%5Cright%29%7D%29%28%5Ccos%7B%5Cleft%28x%20%5Cright%29%7D%29%20" alt="LaTeX:  \displaystyle g'=(\sin{\left(x \right)})(\frac{1}{x})+(\log{\left(x \right)})(\cos{\left(x \right)}) " data-equation-content=" \displaystyle g'=(\sin{\left(x \right)})(\frac{1}{x})+(\log{\left(x \right)})(\cos{\left(x \right)}) " /> Popping up again (Back to the original problem) gives  <img class="equation_image" title=" \displaystyle f'=(7 - 5 x)(\log{\left(x \right)} \cos{\left(x \right)} + \frac{\sin{\left(x \right)}}{x})+(\log{\left(x \right)} \sin{\left(x \right)})(-5)=\left(7 - 5 x\right) \log{\left(x \right)} \cos{\left(x \right)} - 5 \log{\left(x \right)} \sin{\left(x \right)} + \frac{\left(7 - 5 x\right) \sin{\left(x \right)}}{x} " src="/equation_images/%20%5Cdisplaystyle%20f%27%3D%287%20-%205%20x%29%28%5Clog%7B%5Cleft%28x%20%5Cright%29%7D%20%5Ccos%7B%5Cleft%28x%20%5Cright%29%7D%20%2B%20%5Cfrac%7B%5Csin%7B%5Cleft%28x%20%5Cright%29%7D%7D%7Bx%7D%29%2B%28%5Clog%7B%5Cleft%28x%20%5Cright%29%7D%20%5Csin%7B%5Cleft%28x%20%5Cright%29%7D%29%28-5%29%3D%5Cleft%287%20-%205%20x%5Cright%29%20%5Clog%7B%5Cleft%28x%20%5Cright%29%7D%20%5Ccos%7B%5Cleft%28x%20%5Cright%29%7D%20-%205%20%5Clog%7B%5Cleft%28x%20%5Cright%29%7D%20%5Csin%7B%5Cleft%28x%20%5Cright%29%7D%20%2B%20%5Cfrac%7B%5Cleft%287%20-%205%20x%5Cright%29%20%5Csin%7B%5Cleft%28x%20%5Cright%29%7D%7D%7Bx%7D%20" alt="LaTeX:  \displaystyle f'=(7 - 5 x)(\log{\left(x \right)} \cos{\left(x \right)} + \frac{\sin{\left(x \right)}}{x})+(\log{\left(x \right)} \sin{\left(x \right)})(-5)=\left(7 - 5 x\right) \log{\left(x \right)} \cos{\left(x \right)} - 5 \log{\left(x \right)} \sin{\left(x \right)} + \frac{\left(7 - 5 x\right) \sin{\left(x \right)}}{x} " data-equation-content=" \displaystyle f'=(7 - 5 x)(\log{\left(x \right)} \cos{\left(x \right)} + \frac{\sin{\left(x \right)}}{x})+(\log{\left(x \right)} \sin{\left(x \right)})(-5)=\left(7 - 5 x\right) \log{\left(x \right)} \cos{\left(x \right)} - 5 \log{\left(x \right)} \sin{\left(x \right)} + \frac{\left(7 - 5 x\right) \sin{\left(x \right)}}{x} " /> </p> </p>