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Questions: Algebra BusinessCalculus
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Use implicit differentiation where \(\displaystyle y\) is a function of \(\displaystyle x\) to find the derivative of \(\displaystyle 3 e^{x^{2}} \cos{\left(y^{3} \right)} + 2 e^{y^{2}} \cos{\left(x^{3} \right)}=11\)
Taking the derivative of both sides using implicit differentiation gives: \begin{equation*} - 6 x^{2} e^{y^{2}} \sin{\left(x^{3} \right)} + 6 x e^{x^{2}} \cos{\left(y^{3} \right)} - 9 y^{2} y' e^{x^{2}} \sin{\left(y^{3} \right)} + 4 y y' e^{y^{2}} \cos{\left(x^{3} \right)} = 0 \end{equation*} Solving for \(\displaystyle y'\) gives \(\displaystyle y' = \frac{6 x \left(- x e^{y^{2}} \sin{\left(x^{3} \right)} + e^{x^{2}} \cos{\left(y^{3} \right)}\right)}{y \left(9 y e^{x^{2}} \sin{\left(y^{3} \right)} - 4 e^{y^{2}} \cos{\left(x^{3} \right)}\right)}\)
\begin{question}Use implicit differentiation where $y$ is a function of $x$ to find the derivative of $3 e^{x^{2}} \cos{\left(y^{3} \right)} + 2 e^{y^{2}} \cos{\left(x^{3} \right)}=11$
\soln{9cm}{Taking the derivative of both sides using implicit differentiation gives:
\begin{equation*} - 6 x^{2} e^{y^{2}} \sin{\left(x^{3} \right)} + 6 x e^{x^{2}} \cos{\left(y^{3} \right)} - 9 y^{2} y' e^{x^{2}} \sin{\left(y^{3} \right)} + 4 y y' e^{y^{2}} \cos{\left(x^{3} \right)} = 0 \end{equation*}
Solving for $y'$ gives $y' = \frac{6 x \left(- x e^{y^{2}} \sin{\left(x^{3} \right)} + e^{x^{2}} \cos{\left(y^{3} \right)}\right)}{y \left(9 y e^{x^{2}} \sin{\left(y^{3} \right)} - 4 e^{y^{2}} \cos{\left(x^{3} \right)}\right)}$}
\end{question}
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\begin{document}\begin{question}(10pts) The question goes here!
\soln{9cm}{The solution goes here.}
\end{question}\end{document}<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 3 e^{x^{2}} \cos{\left(y^{3} \right)} + 2 e^{y^{2}} \cos{\left(x^{3} \right)}=11 " src="/equation_images/%20%5Cdisplaystyle%203%20e%5E%7Bx%5E%7B2%7D%7D%20%5Ccos%7B%5Cleft%28y%5E%7B3%7D%20%5Cright%29%7D%20%2B%202%20e%5E%7By%5E%7B2%7D%7D%20%5Ccos%7B%5Cleft%28x%5E%7B3%7D%20%5Cright%29%7D%3D11%20" alt="LaTeX: \displaystyle 3 e^{x^{2}} \cos{\left(y^{3} \right)} + 2 e^{y^{2}} \cos{\left(x^{3} \right)}=11 " data-equation-content=" \displaystyle 3 e^{x^{2}} \cos{\left(y^{3} \right)} + 2 e^{y^{2}} \cos{\left(x^{3} \right)}=11 " /> </p> </p><p> <p>Taking the derivative of both sides using implicit differentiation gives:
<img class="equation_image" title=" - 6 x^{2} e^{y^{2}} \sin{\left(x^{3} \right)} + 6 x e^{x^{2}} \cos{\left(y^{3} \right)} - 9 y^{2} y' e^{x^{2}} \sin{\left(y^{3} \right)} + 4 y y' e^{y^{2}} \cos{\left(x^{3} \right)} = 0 " src="/equation_images/%20%20-%206%20x%5E%7B2%7D%20e%5E%7By%5E%7B2%7D%7D%20%5Csin%7B%5Cleft%28x%5E%7B3%7D%20%5Cright%29%7D%20%2B%206%20x%20e%5E%7Bx%5E%7B2%7D%7D%20%5Ccos%7B%5Cleft%28y%5E%7B3%7D%20%5Cright%29%7D%20-%209%20y%5E%7B2%7D%20y%27%20e%5E%7Bx%5E%7B2%7D%7D%20%5Csin%7B%5Cleft%28y%5E%7B3%7D%20%5Cright%29%7D%20%2B%204%20y%20y%27%20e%5E%7By%5E%7B2%7D%7D%20%5Ccos%7B%5Cleft%28x%5E%7B3%7D%20%5Cright%29%7D%20%3D%200%20%20" alt="LaTeX: - 6 x^{2} e^{y^{2}} \sin{\left(x^{3} \right)} + 6 x e^{x^{2}} \cos{\left(y^{3} \right)} - 9 y^{2} y' e^{x^{2}} \sin{\left(y^{3} \right)} + 4 y y' e^{y^{2}} \cos{\left(x^{3} \right)} = 0 " data-equation-content=" - 6 x^{2} e^{y^{2}} \sin{\left(x^{3} \right)} + 6 x e^{x^{2}} \cos{\left(y^{3} \right)} - 9 y^{2} y' e^{x^{2}} \sin{\left(y^{3} \right)} + 4 y y' e^{y^{2}} \cos{\left(x^{3} \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{6 x \left(- x e^{y^{2}} \sin{\left(x^{3} \right)} + e^{x^{2}} \cos{\left(y^{3} \right)}\right)}{y \left(9 y e^{x^{2}} \sin{\left(y^{3} \right)} - 4 e^{y^{2}} \cos{\left(x^{3} \right)}\right)} " src="/equation_images/%20%5Cdisplaystyle%20y%27%20%3D%20%5Cfrac%7B6%20x%20%5Cleft%28-%20x%20e%5E%7By%5E%7B2%7D%7D%20%5Csin%7B%5Cleft%28x%5E%7B3%7D%20%5Cright%29%7D%20%2B%20e%5E%7Bx%5E%7B2%7D%7D%20%5Ccos%7B%5Cleft%28y%5E%7B3%7D%20%5Cright%29%7D%5Cright%29%7D%7By%20%5Cleft%289%20y%20e%5E%7Bx%5E%7B2%7D%7D%20%5Csin%7B%5Cleft%28y%5E%7B3%7D%20%5Cright%29%7D%20-%204%20e%5E%7By%5E%7B2%7D%7D%20%5Ccos%7B%5Cleft%28x%5E%7B3%7D%20%5Cright%29%7D%5Cright%29%7D%20" alt="LaTeX: \displaystyle y' = \frac{6 x \left(- x e^{y^{2}} \sin{\left(x^{3} \right)} + e^{x^{2}} \cos{\left(y^{3} \right)}\right)}{y \left(9 y e^{x^{2}} \sin{\left(y^{3} \right)} - 4 e^{y^{2}} \cos{\left(x^{3} \right)}\right)} " data-equation-content=" \displaystyle y' = \frac{6 x \left(- x e^{y^{2}} \sin{\left(x^{3} \right)} + e^{x^{2}} \cos{\left(y^{3} \right)}\right)}{y \left(9 y e^{x^{2}} \sin{\left(y^{3} \right)} - 4 e^{y^{2}} \cos{\left(x^{3} \right)}\right)} " /> </p> </p>