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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 \(\displaystyle - x^{3} \cos{\left(y^{2} \right)} + 9 \sin{\left(y^{3} \right)} \cos{\left(x^{2} \right)}=36\)
Taking the derivative of both sides using implicit differentiation gives: \begin{equation*} 2 x^{3} y y' \sin{\left(y^{2} \right)} - 3 x^{2} \cos{\left(y^{2} \right)} - 18 x \sin{\left(x^{2} \right)} \sin{\left(y^{3} \right)} + 27 y^{2} y' \cos{\left(x^{2} \right)} \cos{\left(y^{3} \right)} = 0 \end{equation*} Solving for \(\displaystyle y'\) gives \(\displaystyle y' = \frac{3 x \left(x \cos{\left(y^{2} \right)} + 6 \sin{\left(x^{2} \right)} \sin{\left(y^{3} \right)}\right)}{y \left(2 x^{3} \sin{\left(y^{2} \right)} + 27 y \cos{\left(x^{2} \right)} \cos{\left(y^{3} \right)}\right)}\)
\begin{question}Use implicit differentiation where $y$ is a function of $x$ to find the derivative of $- x^{3} \cos{\left(y^{2} \right)} + 9 \sin{\left(y^{3} \right)} \cos{\left(x^{2} \right)}=36$
\soln{9cm}{Taking the derivative of both sides using implicit differentiation gives:
\begin{equation*} 2 x^{3} y y' \sin{\left(y^{2} \right)} - 3 x^{2} \cos{\left(y^{2} \right)} - 18 x \sin{\left(x^{2} \right)} \sin{\left(y^{3} \right)} + 27 y^{2} y' \cos{\left(x^{2} \right)} \cos{\left(y^{3} \right)} = 0 \end{equation*}
Solving for $y'$ gives $y' = \frac{3 x \left(x \cos{\left(y^{2} \right)} + 6 \sin{\left(x^{2} \right)} \sin{\left(y^{3} \right)}\right)}{y \left(2 x^{3} \sin{\left(y^{2} \right)} + 27 y \cos{\left(x^{2} \right)} \cos{\left(y^{3} \right)}\right)}$}
\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 - x^{3} \cos{\left(y^{2} \right)} + 9 \sin{\left(y^{3} \right)} \cos{\left(x^{2} \right)}=36 " src="/equation_images/%20%5Cdisplaystyle%20-%20x%5E%7B3%7D%20%5Ccos%7B%5Cleft%28y%5E%7B2%7D%20%5Cright%29%7D%20%2B%209%20%5Csin%7B%5Cleft%28y%5E%7B3%7D%20%5Cright%29%7D%20%5Ccos%7B%5Cleft%28x%5E%7B2%7D%20%5Cright%29%7D%3D36%20" alt="LaTeX: \displaystyle - x^{3} \cos{\left(y^{2} \right)} + 9 \sin{\left(y^{3} \right)} \cos{\left(x^{2} \right)}=36 " data-equation-content=" \displaystyle - x^{3} \cos{\left(y^{2} \right)} + 9 \sin{\left(y^{3} \right)} \cos{\left(x^{2} \right)}=36 " /> </p> </p><p> <p>Taking the derivative of both sides using implicit differentiation gives:
<img class="equation_image" title=" 2 x^{3} y y' \sin{\left(y^{2} \right)} - 3 x^{2} \cos{\left(y^{2} \right)} - 18 x \sin{\left(x^{2} \right)} \sin{\left(y^{3} \right)} + 27 y^{2} y' \cos{\left(x^{2} \right)} \cos{\left(y^{3} \right)} = 0 " src="/equation_images/%20%202%20x%5E%7B3%7D%20y%20y%27%20%5Csin%7B%5Cleft%28y%5E%7B2%7D%20%5Cright%29%7D%20-%203%20x%5E%7B2%7D%20%5Ccos%7B%5Cleft%28y%5E%7B2%7D%20%5Cright%29%7D%20-%2018%20x%20%5Csin%7B%5Cleft%28x%5E%7B2%7D%20%5Cright%29%7D%20%5Csin%7B%5Cleft%28y%5E%7B3%7D%20%5Cright%29%7D%20%2B%2027%20y%5E%7B2%7D%20y%27%20%5Ccos%7B%5Cleft%28x%5E%7B2%7D%20%5Cright%29%7D%20%5Ccos%7B%5Cleft%28y%5E%7B3%7D%20%5Cright%29%7D%20%3D%200%20%20" alt="LaTeX: 2 x^{3} y y' \sin{\left(y^{2} \right)} - 3 x^{2} \cos{\left(y^{2} \right)} - 18 x \sin{\left(x^{2} \right)} \sin{\left(y^{3} \right)} + 27 y^{2} y' \cos{\left(x^{2} \right)} \cos{\left(y^{3} \right)} = 0 " data-equation-content=" 2 x^{3} y y' \sin{\left(y^{2} \right)} - 3 x^{2} \cos{\left(y^{2} \right)} - 18 x \sin{\left(x^{2} \right)} \sin{\left(y^{3} \right)} + 27 y^{2} y' \cos{\left(x^{2} \right)} \cos{\left(y^{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{3 x \left(x \cos{\left(y^{2} \right)} + 6 \sin{\left(x^{2} \right)} \sin{\left(y^{3} \right)}\right)}{y \left(2 x^{3} \sin{\left(y^{2} \right)} + 27 y \cos{\left(x^{2} \right)} \cos{\left(y^{3} \right)}\right)} " src="/equation_images/%20%5Cdisplaystyle%20y%27%20%3D%20%5Cfrac%7B3%20x%20%5Cleft%28x%20%5Ccos%7B%5Cleft%28y%5E%7B2%7D%20%5Cright%29%7D%20%2B%206%20%5Csin%7B%5Cleft%28x%5E%7B2%7D%20%5Cright%29%7D%20%5Csin%7B%5Cleft%28y%5E%7B3%7D%20%5Cright%29%7D%5Cright%29%7D%7By%20%5Cleft%282%20x%5E%7B3%7D%20%5Csin%7B%5Cleft%28y%5E%7B2%7D%20%5Cright%29%7D%20%2B%2027%20y%20%5Ccos%7B%5Cleft%28x%5E%7B2%7D%20%5Cright%29%7D%20%5Ccos%7B%5Cleft%28y%5E%7B3%7D%20%5Cright%29%7D%5Cright%29%7D%20" alt="LaTeX: \displaystyle y' = \frac{3 x \left(x \cos{\left(y^{2} \right)} + 6 \sin{\left(x^{2} \right)} \sin{\left(y^{3} \right)}\right)}{y \left(2 x^{3} \sin{\left(y^{2} \right)} + 27 y \cos{\left(x^{2} \right)} \cos{\left(y^{3} \right)}\right)} " data-equation-content=" \displaystyle y' = \frac{3 x \left(x \cos{\left(y^{2} \right)} + 6 \sin{\left(x^{2} \right)} \sin{\left(y^{3} \right)}\right)}{y \left(2 x^{3} \sin{\left(y^{2} \right)} + 27 y \cos{\left(x^{2} \right)} \cos{\left(y^{3} \right)}\right)} " /> </p> </p>