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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 \(\displaystyle 14 \sqrt{2} \sqrt{x} \sin{\left(y^{2} \right)} - y^{2} \sin{\left(x^{2} \right)}=-29\)

Taking the derivative of both sides using implicit differentiation gives: \begin{equation*} 28 \sqrt{2} \sqrt{x} y y' \cos{\left(y^{2} \right)} - 2 x y^{2} \cos{\left(x^{2} \right)} - 2 y y' \sin{\left(x^{2} \right)} + \frac{7 \sqrt{2} \sin{\left(y^{2} \right)}}{\sqrt{x}} = 0 \end{equation*} Solving for \(\displaystyle y'\) gives \(\displaystyle y' = \frac{- x^{\frac{3}{2}} y^{2} \cos{\left(x^{2} \right)} + \frac{7 \sqrt{2} \sin{\left(y^{2} \right)}}{2}}{y \left(\sqrt{x} \sin{\left(x^{2} \right)} - 14 \sqrt{2} x \cos{\left(y^{2} \right)}\right)}\)

Download \(\LaTeX\) 

\begin{question}Use implicit differentiation where $y$ is a function of $x$ to find the derivative of $14 \sqrt{2} \sqrt{x} \sin{\left(y^{2} \right)} - y^{2} \sin{\left(x^{2} \right)}=-29$
    \soln{9cm}{Taking the derivative of both sides using implicit differentiation gives:
\begin{equation*} 28 \sqrt{2} \sqrt{x} y y' \cos{\left(y^{2} \right)} - 2 x y^{2} \cos{\left(x^{2} \right)} - 2 y y' \sin{\left(x^{2} \right)} + \frac{7 \sqrt{2} \sin{\left(y^{2} \right)}}{\sqrt{x}} = 0 \end{equation*}
Solving for $y'$ gives $y' = \frac{- x^{\frac{3}{2}} y^{2} \cos{\left(x^{2} \right)} + \frac{7 \sqrt{2} \sin{\left(y^{2} \right)}}{2}}{y \left(\sqrt{x} \sin{\left(x^{2} \right)} - 14 \sqrt{2} x \cos{\left(y^{2} \right)}\right)}$}

\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 14 \sqrt{2} \sqrt{x} \sin{\left(y^{2} \right)} - y^{2} \sin{\left(x^{2} \right)}=-29 " src="/equation_images/%20%5Cdisplaystyle%2014%20%5Csqrt%7B2%7D%20%5Csqrt%7Bx%7D%20%5Csin%7B%5Cleft%28y%5E%7B2%7D%20%5Cright%29%7D%20-%20y%5E%7B2%7D%20%5Csin%7B%5Cleft%28x%5E%7B2%7D%20%5Cright%29%7D%3D-29%20" alt="LaTeX:  \displaystyle 14 \sqrt{2} \sqrt{x} \sin{\left(y^{2} \right)} - y^{2} \sin{\left(x^{2} \right)}=-29 " data-equation-content=" \displaystyle 14 \sqrt{2} \sqrt{x} \sin{\left(y^{2} \right)} - y^{2} \sin{\left(x^{2} \right)}=-29 " /> </p> </p>
HTML for Canvas
<p> <p>Taking the derivative of both sides using implicit differentiation gives:
 <img class="equation_image" title="  28 \sqrt{2} \sqrt{x} y y' \cos{\left(y^{2} \right)} - 2 x y^{2} \cos{\left(x^{2} \right)} - 2 y y' \sin{\left(x^{2} \right)} + \frac{7 \sqrt{2} \sin{\left(y^{2} \right)}}{\sqrt{x}} = 0  " src="/equation_images/%20%2028%20%5Csqrt%7B2%7D%20%5Csqrt%7Bx%7D%20y%20y%27%20%5Ccos%7B%5Cleft%28y%5E%7B2%7D%20%5Cright%29%7D%20-%202%20x%20y%5E%7B2%7D%20%5Ccos%7B%5Cleft%28x%5E%7B2%7D%20%5Cright%29%7D%20-%202%20y%20y%27%20%5Csin%7B%5Cleft%28x%5E%7B2%7D%20%5Cright%29%7D%20%2B%20%5Cfrac%7B7%20%5Csqrt%7B2%7D%20%5Csin%7B%5Cleft%28y%5E%7B2%7D%20%5Cright%29%7D%7D%7B%5Csqrt%7Bx%7D%7D%20%3D%200%20%20" alt="LaTeX:   28 \sqrt{2} \sqrt{x} y y' \cos{\left(y^{2} \right)} - 2 x y^{2} \cos{\left(x^{2} \right)} - 2 y y' \sin{\left(x^{2} \right)} + \frac{7 \sqrt{2} \sin{\left(y^{2} \right)}}{\sqrt{x}} = 0  " data-equation-content="  28 \sqrt{2} \sqrt{x} y y' \cos{\left(y^{2} \right)} - 2 x y^{2} \cos{\left(x^{2} \right)} - 2 y y' \sin{\left(x^{2} \right)} + \frac{7 \sqrt{2} \sin{\left(y^{2} \right)}}{\sqrt{x}} = 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{- x^{\frac{3}{2}} y^{2} \cos{\left(x^{2} \right)} + \frac{7 \sqrt{2} \sin{\left(y^{2} \right)}}{2}}{y \left(\sqrt{x} \sin{\left(x^{2} \right)} - 14 \sqrt{2} x \cos{\left(y^{2} \right)}\right)} " src="/equation_images/%20%5Cdisplaystyle%20y%27%20%3D%20%5Cfrac%7B-%20x%5E%7B%5Cfrac%7B3%7D%7B2%7D%7D%20y%5E%7B2%7D%20%5Ccos%7B%5Cleft%28x%5E%7B2%7D%20%5Cright%29%7D%20%2B%20%5Cfrac%7B7%20%5Csqrt%7B2%7D%20%5Csin%7B%5Cleft%28y%5E%7B2%7D%20%5Cright%29%7D%7D%7B2%7D%7D%7By%20%5Cleft%28%5Csqrt%7Bx%7D%20%5Csin%7B%5Cleft%28x%5E%7B2%7D%20%5Cright%29%7D%20-%2014%20%5Csqrt%7B2%7D%20x%20%5Ccos%7B%5Cleft%28y%5E%7B2%7D%20%5Cright%29%7D%5Cright%29%7D%20" alt="LaTeX:  \displaystyle y' = \frac{- x^{\frac{3}{2}} y^{2} \cos{\left(x^{2} \right)} + \frac{7 \sqrt{2} \sin{\left(y^{2} \right)}}{2}}{y \left(\sqrt{x} \sin{\left(x^{2} \right)} - 14 \sqrt{2} x \cos{\left(y^{2} \right)}\right)} " data-equation-content=" \displaystyle y' = \frac{- x^{\frac{3}{2}} y^{2} \cos{\left(x^{2} \right)} + \frac{7 \sqrt{2} \sin{\left(y^{2} \right)}}{2}}{y \left(\sqrt{x} \sin{\left(x^{2} \right)} - 14 \sqrt{2} x \cos{\left(y^{2} \right)}\right)} " /> </p> </p>