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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 \(\displaystyle - 9 \sqrt{6} \sqrt{x} \sin{\left(y \right)} + 5 y^{2} \sin{\left(x \right)}=7\)
Taking the derivative of both sides using implicit differentiation gives: \begin{equation*} - 9 \sqrt{6} \sqrt{x} y' \cos{\left(y \right)} + 5 y^{2} \cos{\left(x \right)} + 10 y y' \sin{\left(x \right)} - \frac{9 \sqrt{6} \sin{\left(y \right)}}{2 \sqrt{x}} = 0 \end{equation*} Solving for \(\displaystyle y'\) gives \(\displaystyle y' = \frac{- 10 \sqrt{x} y^{2} \cos{\left(x \right)} + 9 \sqrt{6} \sin{\left(y \right)}}{2 \left(10 \sqrt{x} y \sin{\left(x \right)} - 9 \sqrt{6} x \cos{\left(y \right)}\right)}\)
\begin{question}Use implicit differentiation where $y$ is a function of $x$ to find the derivative of $- 9 \sqrt{6} \sqrt{x} \sin{\left(y \right)} + 5 y^{2} \sin{\left(x \right)}=7$
\soln{9cm}{Taking the derivative of both sides using implicit differentiation gives:
\begin{equation*} - 9 \sqrt{6} \sqrt{x} y' \cos{\left(y \right)} + 5 y^{2} \cos{\left(x \right)} + 10 y y' \sin{\left(x \right)} - \frac{9 \sqrt{6} \sin{\left(y \right)}}{2 \sqrt{x}} = 0 \end{equation*}
Solving for $y'$ gives $y' = \frac{- 10 \sqrt{x} y^{2} \cos{\left(x \right)} + 9 \sqrt{6} \sin{\left(y \right)}}{2 \left(10 \sqrt{x} y \sin{\left(x \right)} - 9 \sqrt{6} x \cos{\left(y \right)}\right)}$}
\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 - 9 \sqrt{6} \sqrt{x} \sin{\left(y \right)} + 5 y^{2} \sin{\left(x \right)}=7 " src="/equation_images/%20%5Cdisplaystyle%20-%209%20%5Csqrt%7B6%7D%20%5Csqrt%7Bx%7D%20%5Csin%7B%5Cleft%28y%20%5Cright%29%7D%20%2B%205%20y%5E%7B2%7D%20%5Csin%7B%5Cleft%28x%20%5Cright%29%7D%3D7%20" alt="LaTeX: \displaystyle - 9 \sqrt{6} \sqrt{x} \sin{\left(y \right)} + 5 y^{2} \sin{\left(x \right)}=7 " data-equation-content=" \displaystyle - 9 \sqrt{6} \sqrt{x} \sin{\left(y \right)} + 5 y^{2} \sin{\left(x \right)}=7 " /> </p> </p><p> <p>Taking the derivative of both sides using implicit differentiation gives:
<img class="equation_image" title=" - 9 \sqrt{6} \sqrt{x} y' \cos{\left(y \right)} + 5 y^{2} \cos{\left(x \right)} + 10 y y' \sin{\left(x \right)} - \frac{9 \sqrt{6} \sin{\left(y \right)}}{2 \sqrt{x}} = 0 " src="/equation_images/%20%20-%209%20%5Csqrt%7B6%7D%20%5Csqrt%7Bx%7D%20y%27%20%5Ccos%7B%5Cleft%28y%20%5Cright%29%7D%20%2B%205%20y%5E%7B2%7D%20%5Ccos%7B%5Cleft%28x%20%5Cright%29%7D%20%2B%2010%20y%20y%27%20%5Csin%7B%5Cleft%28x%20%5Cright%29%7D%20-%20%5Cfrac%7B9%20%5Csqrt%7B6%7D%20%5Csin%7B%5Cleft%28y%20%5Cright%29%7D%7D%7B2%20%5Csqrt%7Bx%7D%7D%20%3D%200%20%20" alt="LaTeX: - 9 \sqrt{6} \sqrt{x} y' \cos{\left(y \right)} + 5 y^{2} \cos{\left(x \right)} + 10 y y' \sin{\left(x \right)} - \frac{9 \sqrt{6} \sin{\left(y \right)}}{2 \sqrt{x}} = 0 " data-equation-content=" - 9 \sqrt{6} \sqrt{x} y' \cos{\left(y \right)} + 5 y^{2} \cos{\left(x \right)} + 10 y y' \sin{\left(x \right)} - \frac{9 \sqrt{6} \sin{\left(y \right)}}{2 \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{- 10 \sqrt{x} y^{2} \cos{\left(x \right)} + 9 \sqrt{6} \sin{\left(y \right)}}{2 \left(10 \sqrt{x} y \sin{\left(x \right)} - 9 \sqrt{6} x \cos{\left(y \right)}\right)} " src="/equation_images/%20%5Cdisplaystyle%20y%27%20%3D%20%5Cfrac%7B-%2010%20%5Csqrt%7Bx%7D%20y%5E%7B2%7D%20%5Ccos%7B%5Cleft%28x%20%5Cright%29%7D%20%2B%209%20%5Csqrt%7B6%7D%20%5Csin%7B%5Cleft%28y%20%5Cright%29%7D%7D%7B2%20%5Cleft%2810%20%5Csqrt%7Bx%7D%20y%20%5Csin%7B%5Cleft%28x%20%5Cright%29%7D%20-%209%20%5Csqrt%7B6%7D%20x%20%5Ccos%7B%5Cleft%28y%20%5Cright%29%7D%5Cright%29%7D%20" alt="LaTeX: \displaystyle y' = \frac{- 10 \sqrt{x} y^{2} \cos{\left(x \right)} + 9 \sqrt{6} \sin{\left(y \right)}}{2 \left(10 \sqrt{x} y \sin{\left(x \right)} - 9 \sqrt{6} x \cos{\left(y \right)}\right)} " data-equation-content=" \displaystyle y' = \frac{- 10 \sqrt{x} y^{2} \cos{\left(x \right)} + 9 \sqrt{6} \sin{\left(y \right)}}{2 \left(10 \sqrt{x} y \sin{\left(x \right)} - 9 \sqrt{6} x \cos{\left(y \right)}\right)} " /> </p> </p>