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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 \(\displaystyle 5 \sqrt{7} \sqrt{y} e^{x^{2}} + 3 e^{y^{2}} \sin{\left(x^{3} \right)}=30\)

Taking the derivative of both sides using implicit differentiation gives: \begin{equation*} 9 x^{2} e^{y^{2}} \cos{\left(x^{3} \right)} + 10 \sqrt{7} x \sqrt{y} e^{x^{2}} + 6 y y' e^{y^{2}} \sin{\left(x^{3} \right)} + \frac{5 \sqrt{7} y' e^{x^{2}}}{2 \sqrt{y}} = 0 \end{equation*} Solving for \(\displaystyle y'\) gives \(\displaystyle y' = - \frac{2 x \left(9 x \sqrt{y} e^{y^{2}} \cos{\left(x^{3} \right)} + 10 \sqrt{7} y e^{x^{2}}\right)}{12 y^{\frac{3}{2}} e^{y^{2}} \sin{\left(x^{3} \right)} + 5 \sqrt{7} e^{x^{2}}}\)

Download \(\LaTeX\) 

\begin{question}Use implicit differentiation where $y$ is a function of $x$ to find the derivative of $5 \sqrt{7} \sqrt{y} e^{x^{2}} + 3 e^{y^{2}} \sin{\left(x^{3} \right)}=30$
    \soln{9cm}{Taking the derivative of both sides using implicit differentiation gives:
\begin{equation*} 9 x^{2} e^{y^{2}} \cos{\left(x^{3} \right)} + 10 \sqrt{7} x \sqrt{y} e^{x^{2}} + 6 y y' e^{y^{2}} \sin{\left(x^{3} \right)} + \frac{5 \sqrt{7} y' e^{x^{2}}}{2 \sqrt{y}} = 0 \end{equation*}
Solving for $y'$ gives $y' = - \frac{2 x \left(9 x \sqrt{y} e^{y^{2}} \cos{\left(x^{3} \right)} + 10 \sqrt{7} y e^{x^{2}}\right)}{12 y^{\frac{3}{2}} e^{y^{2}} \sin{\left(x^{3} \right)} + 5 \sqrt{7} e^{x^{2}}}$}

\end{question}

Download Question and Solution Environment\(\LaTeX\)
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    \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 5 \sqrt{7} \sqrt{y} e^{x^{2}} + 3 e^{y^{2}} \sin{\left(x^{3} \right)}=30 " src="/equation_images/%20%5Cdisplaystyle%205%20%5Csqrt%7B7%7D%20%5Csqrt%7By%7D%20e%5E%7Bx%5E%7B2%7D%7D%20%2B%203%20e%5E%7By%5E%7B2%7D%7D%20%5Csin%7B%5Cleft%28x%5E%7B3%7D%20%5Cright%29%7D%3D30%20" alt="LaTeX:  \displaystyle 5 \sqrt{7} \sqrt{y} e^{x^{2}} + 3 e^{y^{2}} \sin{\left(x^{3} \right)}=30 " data-equation-content=" \displaystyle 5 \sqrt{7} \sqrt{y} e^{x^{2}} + 3 e^{y^{2}} \sin{\left(x^{3} \right)}=30 " /> </p> </p>
HTML for Canvas
<p> <p>Taking the derivative of both sides using implicit differentiation gives:
 <img class="equation_image" title="  9 x^{2} e^{y^{2}} \cos{\left(x^{3} \right)} + 10 \sqrt{7} x \sqrt{y} e^{x^{2}} + 6 y y' e^{y^{2}} \sin{\left(x^{3} \right)} + \frac{5 \sqrt{7} y' e^{x^{2}}}{2 \sqrt{y}} = 0  " src="/equation_images/%20%209%20x%5E%7B2%7D%20e%5E%7By%5E%7B2%7D%7D%20%5Ccos%7B%5Cleft%28x%5E%7B3%7D%20%5Cright%29%7D%20%2B%2010%20%5Csqrt%7B7%7D%20x%20%5Csqrt%7By%7D%20e%5E%7Bx%5E%7B2%7D%7D%20%2B%206%20y%20y%27%20e%5E%7By%5E%7B2%7D%7D%20%5Csin%7B%5Cleft%28x%5E%7B3%7D%20%5Cright%29%7D%20%2B%20%5Cfrac%7B5%20%5Csqrt%7B7%7D%20y%27%20e%5E%7Bx%5E%7B2%7D%7D%7D%7B2%20%5Csqrt%7By%7D%7D%20%3D%200%20%20" alt="LaTeX:   9 x^{2} e^{y^{2}} \cos{\left(x^{3} \right)} + 10 \sqrt{7} x \sqrt{y} e^{x^{2}} + 6 y y' e^{y^{2}} \sin{\left(x^{3} \right)} + \frac{5 \sqrt{7} y' e^{x^{2}}}{2 \sqrt{y}} = 0  " data-equation-content="  9 x^{2} e^{y^{2}} \cos{\left(x^{3} \right)} + 10 \sqrt{7} x \sqrt{y} e^{x^{2}} + 6 y y' e^{y^{2}} \sin{\left(x^{3} \right)} + \frac{5 \sqrt{7} y' e^{x^{2}}}{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 x \left(9 x \sqrt{y} e^{y^{2}} \cos{\left(x^{3} \right)} + 10 \sqrt{7} y e^{x^{2}}\right)}{12 y^{\frac{3}{2}} e^{y^{2}} \sin{\left(x^{3} \right)} + 5 \sqrt{7} e^{x^{2}}} " src="/equation_images/%20%5Cdisplaystyle%20y%27%20%3D%20-%20%5Cfrac%7B2%20x%20%5Cleft%289%20x%20%5Csqrt%7By%7D%20e%5E%7By%5E%7B2%7D%7D%20%5Ccos%7B%5Cleft%28x%5E%7B3%7D%20%5Cright%29%7D%20%2B%2010%20%5Csqrt%7B7%7D%20y%20e%5E%7Bx%5E%7B2%7D%7D%5Cright%29%7D%7B12%20y%5E%7B%5Cfrac%7B3%7D%7B2%7D%7D%20e%5E%7By%5E%7B2%7D%7D%20%5Csin%7B%5Cleft%28x%5E%7B3%7D%20%5Cright%29%7D%20%2B%205%20%5Csqrt%7B7%7D%20e%5E%7Bx%5E%7B2%7D%7D%7D%20" alt="LaTeX:  \displaystyle y' = - \frac{2 x \left(9 x \sqrt{y} e^{y^{2}} \cos{\left(x^{3} \right)} + 10 \sqrt{7} y e^{x^{2}}\right)}{12 y^{\frac{3}{2}} e^{y^{2}} \sin{\left(x^{3} \right)} + 5 \sqrt{7} e^{x^{2}}} " data-equation-content=" \displaystyle y' = - \frac{2 x \left(9 x \sqrt{y} e^{y^{2}} \cos{\left(x^{3} \right)} + 10 \sqrt{7} y e^{x^{2}}\right)}{12 y^{\frac{3}{2}} e^{y^{2}} \sin{\left(x^{3} \right)} + 5 \sqrt{7} e^{x^{2}}} " /> </p> </p>