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Calculus
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Use implicit differentiation where \(\displaystyle y\) is a function of \(\displaystyle x\) to find the derivative of \(\displaystyle 7 e^{x^{3}} \cos{\left(y \right)} - 5 \sin{\left(x \right)} \sin{\left(y \right)}=-41\)

Taking the derivative of both sides using implicit differentiation gives: \begin{equation*} 21 x^{2} e^{x^{3}} \cos{\left(y \right)} - 7 y' e^{x^{3}} \sin{\left(y \right)} - 5 y' \sin{\left(x \right)} \cos{\left(y \right)} - 5 \sin{\left(y \right)} \cos{\left(x \right)} = 0 \end{equation*} Solving for \(\displaystyle y'\) gives \(\displaystyle y' = \frac{21 x^{2} e^{x^{3}} \cos{\left(y \right)} - 5 \sin{\left(y \right)} \cos{\left(x \right)}}{7 e^{x^{3}} \sin{\left(y \right)} + 5 \sin{\left(x \right)} \cos{\left(y \right)}}\)

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\begin{question}Use implicit differentiation where $y$ is a function of $x$ to find the derivative of $7 e^{x^{3}} \cos{\left(y \right)} - 5 \sin{\left(x \right)} \sin{\left(y \right)}=-41$
    \soln{9cm}{Taking the derivative of both sides using implicit differentiation gives:
\begin{equation*} 21 x^{2} e^{x^{3}} \cos{\left(y \right)} - 7 y' e^{x^{3}} \sin{\left(y \right)} - 5 y' \sin{\left(x \right)} \cos{\left(y \right)} - 5 \sin{\left(y \right)} \cos{\left(x \right)} = 0 \end{equation*}
Solving for $y'$ gives $y' = \frac{21 x^{2} e^{x^{3}} \cos{\left(y \right)} - 5 \sin{\left(y \right)} \cos{\left(x \right)}}{7 e^{x^{3}} \sin{\left(y \right)} + 5 \sin{\left(x \right)} \cos{\left(y \right)}}$}

\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>Use implicit differentiation where  <img class="equation_image" title=" \displaystyle y " src="/equation_images/%20%5Cdisplaystyle%20y%20" alt="LaTeX:  \displaystyle y " data-equation-content=" \displaystyle y " />  is a function of  <img class="equation_image" title=" \displaystyle x " src="/equation_images/%20%5Cdisplaystyle%20x%20" alt="LaTeX:  \displaystyle x " data-equation-content=" \displaystyle x " />  to find the derivative of  <img class="equation_image" title=" \displaystyle 7 e^{x^{3}} \cos{\left(y \right)} - 5 \sin{\left(x \right)} \sin{\left(y \right)}=-41 " src="/equation_images/%20%5Cdisplaystyle%207%20e%5E%7Bx%5E%7B3%7D%7D%20%5Ccos%7B%5Cleft%28y%20%5Cright%29%7D%20-%205%20%5Csin%7B%5Cleft%28x%20%5Cright%29%7D%20%5Csin%7B%5Cleft%28y%20%5Cright%29%7D%3D-41%20" alt="LaTeX:  \displaystyle 7 e^{x^{3}} \cos{\left(y \right)} - 5 \sin{\left(x \right)} \sin{\left(y \right)}=-41 " data-equation-content=" \displaystyle 7 e^{x^{3}} \cos{\left(y \right)} - 5 \sin{\left(x \right)} \sin{\left(y \right)}=-41 " /> </p> </p>
HTML for Canvas
<p> <p>Taking the derivative of both sides using implicit differentiation gives:
 <img class="equation_image" title="  21 x^{2} e^{x^{3}} \cos{\left(y \right)} - 7 y' e^{x^{3}} \sin{\left(y \right)} - 5 y' \sin{\left(x \right)} \cos{\left(y \right)} - 5 \sin{\left(y \right)} \cos{\left(x \right)} = 0  " src="/equation_images/%20%2021%20x%5E%7B2%7D%20e%5E%7Bx%5E%7B3%7D%7D%20%5Ccos%7B%5Cleft%28y%20%5Cright%29%7D%20-%207%20y%27%20e%5E%7Bx%5E%7B3%7D%7D%20%5Csin%7B%5Cleft%28y%20%5Cright%29%7D%20-%205%20y%27%20%5Csin%7B%5Cleft%28x%20%5Cright%29%7D%20%5Ccos%7B%5Cleft%28y%20%5Cright%29%7D%20-%205%20%5Csin%7B%5Cleft%28y%20%5Cright%29%7D%20%5Ccos%7B%5Cleft%28x%20%5Cright%29%7D%20%3D%200%20%20" alt="LaTeX:   21 x^{2} e^{x^{3}} \cos{\left(y \right)} - 7 y' e^{x^{3}} \sin{\left(y \right)} - 5 y' \sin{\left(x \right)} \cos{\left(y \right)} - 5 \sin{\left(y \right)} \cos{\left(x \right)} = 0  " data-equation-content="  21 x^{2} e^{x^{3}} \cos{\left(y \right)} - 7 y' e^{x^{3}} \sin{\left(y \right)} - 5 y' \sin{\left(x \right)} \cos{\left(y \right)} - 5 \sin{\left(y \right)} \cos{\left(x \right)} = 0  " /> 
Solving for  <img class="equation_image" title=" \displaystyle y' " src="/equation_images/%20%5Cdisplaystyle%20y%27%20" alt="LaTeX:  \displaystyle y' " data-equation-content=" \displaystyle y' " />  gives  <img class="equation_image" title=" \displaystyle y' = \frac{21 x^{2} e^{x^{3}} \cos{\left(y \right)} - 5 \sin{\left(y \right)} \cos{\left(x \right)}}{7 e^{x^{3}} \sin{\left(y \right)} + 5 \sin{\left(x \right)} \cos{\left(y \right)}} " src="/equation_images/%20%5Cdisplaystyle%20y%27%20%3D%20%5Cfrac%7B21%20x%5E%7B2%7D%20e%5E%7Bx%5E%7B3%7D%7D%20%5Ccos%7B%5Cleft%28y%20%5Cright%29%7D%20-%205%20%5Csin%7B%5Cleft%28y%20%5Cright%29%7D%20%5Ccos%7B%5Cleft%28x%20%5Cright%29%7D%7D%7B7%20e%5E%7Bx%5E%7B3%7D%7D%20%5Csin%7B%5Cleft%28y%20%5Cright%29%7D%20%2B%205%20%5Csin%7B%5Cleft%28x%20%5Cright%29%7D%20%5Ccos%7B%5Cleft%28y%20%5Cright%29%7D%7D%20" alt="LaTeX:  \displaystyle y' = \frac{21 x^{2} e^{x^{3}} \cos{\left(y \right)} - 5 \sin{\left(y \right)} \cos{\left(x \right)}}{7 e^{x^{3}} \sin{\left(y \right)} + 5 \sin{\left(x \right)} \cos{\left(y \right)}} " data-equation-content=" \displaystyle y' = \frac{21 x^{2} e^{x^{3}} \cos{\left(y \right)} - 5 \sin{\left(y \right)} \cos{\left(x \right)}}{7 e^{x^{3}} \sin{\left(y \right)} + 5 \sin{\left(x \right)} \cos{\left(y \right)}} " /> </p> </p>