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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 \(\displaystyle 4 x^{3} e^{y^{2}} + \sqrt{3} \sqrt{y} \cos{\left(x^{2} \right)}=-14\)
Taking the derivative of both sides using implicit differentiation gives: \begin{equation*} 8 x^{3} y y' e^{y^{2}} + 12 x^{2} e^{y^{2}} - 2 \sqrt{3} x \sqrt{y} \sin{\left(x^{2} \right)} + \frac{\sqrt{3} y' \cos{\left(x^{2} \right)}}{2 \sqrt{y}} = 0 \end{equation*} Solving for \(\displaystyle y'\) gives \(\displaystyle y' = \frac{4 x \left(- 6 x \sqrt{y} e^{y^{2}} + \sqrt{3} y \sin{\left(x^{2} \right)}\right)}{16 x^{3} y^{\frac{3}{2}} e^{y^{2}} + \sqrt{3} \cos{\left(x^{2} \right)}}\)
\begin{question}Use implicit differentiation where $y$ is a function of $x$ to find the derivative of $4 x^{3} e^{y^{2}} + \sqrt{3} \sqrt{y} \cos{\left(x^{2} \right)}=-14$
\soln{9cm}{Taking the derivative of both sides using implicit differentiation gives:
\begin{equation*} 8 x^{3} y y' e^{y^{2}} + 12 x^{2} e^{y^{2}} - 2 \sqrt{3} x \sqrt{y} \sin{\left(x^{2} \right)} + \frac{\sqrt{3} y' \cos{\left(x^{2} \right)}}{2 \sqrt{y}} = 0 \end{equation*}
Solving for $y'$ gives $y' = \frac{4 x \left(- 6 x \sqrt{y} e^{y^{2}} + \sqrt{3} y \sin{\left(x^{2} \right)}\right)}{16 x^{3} y^{\frac{3}{2}} e^{y^{2}} + \sqrt{3} \cos{\left(x^{2} \right)}}$}
\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 4 x^{3} e^{y^{2}} + \sqrt{3} \sqrt{y} \cos{\left(x^{2} \right)}=-14 " src="/equation_images/%20%5Cdisplaystyle%204%20x%5E%7B3%7D%20e%5E%7By%5E%7B2%7D%7D%20%2B%20%5Csqrt%7B3%7D%20%5Csqrt%7By%7D%20%5Ccos%7B%5Cleft%28x%5E%7B2%7D%20%5Cright%29%7D%3D-14%20" alt="LaTeX: \displaystyle 4 x^{3} e^{y^{2}} + \sqrt{3} \sqrt{y} \cos{\left(x^{2} \right)}=-14 " data-equation-content=" \displaystyle 4 x^{3} e^{y^{2}} + \sqrt{3} \sqrt{y} \cos{\left(x^{2} \right)}=-14 " /> </p> </p><p> <p>Taking the derivative of both sides using implicit differentiation gives:
<img class="equation_image" title=" 8 x^{3} y y' e^{y^{2}} + 12 x^{2} e^{y^{2}} - 2 \sqrt{3} x \sqrt{y} \sin{\left(x^{2} \right)} + \frac{\sqrt{3} y' \cos{\left(x^{2} \right)}}{2 \sqrt{y}} = 0 " src="/equation_images/%20%208%20x%5E%7B3%7D%20y%20y%27%20e%5E%7By%5E%7B2%7D%7D%20%2B%2012%20x%5E%7B2%7D%20e%5E%7By%5E%7B2%7D%7D%20-%202%20%5Csqrt%7B3%7D%20x%20%5Csqrt%7By%7D%20%5Csin%7B%5Cleft%28x%5E%7B2%7D%20%5Cright%29%7D%20%2B%20%5Cfrac%7B%5Csqrt%7B3%7D%20y%27%20%5Ccos%7B%5Cleft%28x%5E%7B2%7D%20%5Cright%29%7D%7D%7B2%20%5Csqrt%7By%7D%7D%20%3D%200%20%20" alt="LaTeX: 8 x^{3} y y' e^{y^{2}} + 12 x^{2} e^{y^{2}} - 2 \sqrt{3} x \sqrt{y} \sin{\left(x^{2} \right)} + \frac{\sqrt{3} y' \cos{\left(x^{2} \right)}}{2 \sqrt{y}} = 0 " data-equation-content=" 8 x^{3} y y' e^{y^{2}} + 12 x^{2} e^{y^{2}} - 2 \sqrt{3} x \sqrt{y} \sin{\left(x^{2} \right)} + \frac{\sqrt{3} y' \cos{\left(x^{2} \right)}}{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{4 x \left(- 6 x \sqrt{y} e^{y^{2}} + \sqrt{3} y \sin{\left(x^{2} \right)}\right)}{16 x^{3} y^{\frac{3}{2}} e^{y^{2}} + \sqrt{3} \cos{\left(x^{2} \right)}} " src="/equation_images/%20%5Cdisplaystyle%20y%27%20%3D%20%5Cfrac%7B4%20x%20%5Cleft%28-%206%20x%20%5Csqrt%7By%7D%20e%5E%7By%5E%7B2%7D%7D%20%2B%20%5Csqrt%7B3%7D%20y%20%5Csin%7B%5Cleft%28x%5E%7B2%7D%20%5Cright%29%7D%5Cright%29%7D%7B16%20x%5E%7B3%7D%20y%5E%7B%5Cfrac%7B3%7D%7B2%7D%7D%20e%5E%7By%5E%7B2%7D%7D%20%2B%20%5Csqrt%7B3%7D%20%5Ccos%7B%5Cleft%28x%5E%7B2%7D%20%5Cright%29%7D%7D%20" alt="LaTeX: \displaystyle y' = \frac{4 x \left(- 6 x \sqrt{y} e^{y^{2}} + \sqrt{3} y \sin{\left(x^{2} \right)}\right)}{16 x^{3} y^{\frac{3}{2}} e^{y^{2}} + \sqrt{3} \cos{\left(x^{2} \right)}} " data-equation-content=" \displaystyle y' = \frac{4 x \left(- 6 x \sqrt{y} e^{y^{2}} + \sqrt{3} y \sin{\left(x^{2} \right)}\right)}{16 x^{3} y^{\frac{3}{2}} e^{y^{2}} + \sqrt{3} \cos{\left(x^{2} \right)}} " /> </p> </p>