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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 \(\displaystyle - x^{2} \log{\left(y \right)} + 3 \sqrt{2} \sqrt{y} e^{x}=47\)

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

Download \(\LaTeX\) 

\begin{question}Use implicit differentiation where $y$ is a function of $x$ to find the derivative of $- x^{2} \log{\left(y \right)} + 3 \sqrt{2} \sqrt{y} e^{x}=47$
    \soln{9cm}{Taking the derivative of both sides using implicit differentiation gives:
\begin{equation*} - \frac{x^{2} y'}{y} - 2 x \log{\left(y \right)} + 3 \sqrt{2} \sqrt{y} e^{x} + \frac{3 \sqrt{2} y' e^{x}}{2 \sqrt{y}} = 0 \end{equation*}
Solving for $y'$ gives $y' = \frac{2 \left(- 2 x y^{\frac{3}{2}} \log{\left(y \right)} + 3 \sqrt{2} y^{2} e^{x}\right)}{2 x^{2} \sqrt{y} - 3 \sqrt{2} y e^{x}}$}

\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>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 - x^{2} \log{\left(y \right)} + 3 \sqrt{2} \sqrt{y} e^{x}=47 " src="/equation_images/%20%5Cdisplaystyle%20-%20x%5E%7B2%7D%20%5Clog%7B%5Cleft%28y%20%5Cright%29%7D%20%2B%203%20%5Csqrt%7B2%7D%20%5Csqrt%7By%7D%20e%5E%7Bx%7D%3D47%20" alt="LaTeX:  \displaystyle - x^{2} \log{\left(y \right)} + 3 \sqrt{2} \sqrt{y} e^{x}=47 " data-equation-content=" \displaystyle - x^{2} \log{\left(y \right)} + 3 \sqrt{2} \sqrt{y} e^{x}=47 " /> </p> </p>
HTML for Canvas
<p> <p>Taking the derivative of both sides using implicit differentiation gives:
 <img class="equation_image" title="  - \frac{x^{2} y'}{y} - 2 x \log{\left(y \right)} + 3 \sqrt{2} \sqrt{y} e^{x} + \frac{3 \sqrt{2} y' e^{x}}{2 \sqrt{y}} = 0  " src="/equation_images/%20%20-%20%5Cfrac%7Bx%5E%7B2%7D%20y%27%7D%7By%7D%20-%202%20x%20%5Clog%7B%5Cleft%28y%20%5Cright%29%7D%20%2B%203%20%5Csqrt%7B2%7D%20%5Csqrt%7By%7D%20e%5E%7Bx%7D%20%2B%20%5Cfrac%7B3%20%5Csqrt%7B2%7D%20y%27%20e%5E%7Bx%7D%7D%7B2%20%5Csqrt%7By%7D%7D%20%3D%200%20%20" alt="LaTeX:   - \frac{x^{2} y'}{y} - 2 x \log{\left(y \right)} + 3 \sqrt{2} \sqrt{y} e^{x} + \frac{3 \sqrt{2} y' e^{x}}{2 \sqrt{y}} = 0  " data-equation-content="  - \frac{x^{2} y'}{y} - 2 x \log{\left(y \right)} + 3 \sqrt{2} \sqrt{y} e^{x} + \frac{3 \sqrt{2} y' e^{x}}{2 \sqrt{y}} = 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{2 \left(- 2 x y^{\frac{3}{2}} \log{\left(y \right)} + 3 \sqrt{2} y^{2} e^{x}\right)}{2 x^{2} \sqrt{y} - 3 \sqrt{2} y e^{x}} " src="/equation_images/%20%5Cdisplaystyle%20y%27%20%3D%20%5Cfrac%7B2%20%5Cleft%28-%202%20x%20y%5E%7B%5Cfrac%7B3%7D%7B2%7D%7D%20%5Clog%7B%5Cleft%28y%20%5Cright%29%7D%20%2B%203%20%5Csqrt%7B2%7D%20y%5E%7B2%7D%20e%5E%7Bx%7D%5Cright%29%7D%7B2%20x%5E%7B2%7D%20%5Csqrt%7By%7D%20-%203%20%5Csqrt%7B2%7D%20y%20e%5E%7Bx%7D%7D%20" alt="LaTeX:  \displaystyle y' = \frac{2 \left(- 2 x y^{\frac{3}{2}} \log{\left(y \right)} + 3 \sqrt{2} y^{2} e^{x}\right)}{2 x^{2} \sqrt{y} - 3 \sqrt{2} y e^{x}} " data-equation-content=" \displaystyle y' = \frac{2 \left(- 2 x y^{\frac{3}{2}} \log{\left(y \right)} + 3 \sqrt{2} y^{2} e^{x}\right)}{2 x^{2} \sqrt{y} - 3 \sqrt{2} y e^{x}} " /> </p> </p>