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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 \(\displaystyle 27 x^{2} \sqrt{y} - 5 e^{y} \sin{\left(x \right)}=-5\)

Taking the derivative of both sides using implicit differentiation gives: \begin{equation*} \frac{27 x^{2} y'}{2 \sqrt{y}} + 54 x \sqrt{y} - 5 y' e^{y} \sin{\left(x \right)} - 5 e^{y} \cos{\left(x \right)} = 0 \end{equation*} Solving for \(\displaystyle y'\) gives \(\displaystyle y' = \frac{2 \left(- 54 x y + 5 \sqrt{y} e^{y} \cos{\left(x \right)}\right)}{27 x^{2} - 10 \sqrt{y} e^{y} \sin{\left(x \right)}}\)

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\begin{question}Use implicit differentiation where $y$ is a function of $x$ to find the derivative of $27 x^{2} \sqrt{y} - 5 e^{y} \sin{\left(x \right)}=-5$
    \soln{9cm}{Taking the derivative of both sides using implicit differentiation gives:
\begin{equation*} \frac{27 x^{2} y'}{2 \sqrt{y}} + 54 x \sqrt{y} - 5 y' e^{y} \sin{\left(x \right)} - 5 e^{y} \cos{\left(x \right)} = 0 \end{equation*}
Solving for $y'$ gives $y' = \frac{2 \left(- 54 x y + 5 \sqrt{y} e^{y} \cos{\left(x \right)}\right)}{27 x^{2} - 10 \sqrt{y} e^{y} \sin{\left(x \right)}}$}

\end{question}

Download Question and Solution Environment\(\LaTeX\)
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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 27 x^{2} \sqrt{y} - 5 e^{y} \sin{\left(x \right)}=-5 " src="/equation_images/%20%5Cdisplaystyle%2027%20x%5E%7B2%7D%20%5Csqrt%7By%7D%20-%205%20e%5E%7By%7D%20%5Csin%7B%5Cleft%28x%20%5Cright%29%7D%3D-5%20" alt="LaTeX:  \displaystyle 27 x^{2} \sqrt{y} - 5 e^{y} \sin{\left(x \right)}=-5 " data-equation-content=" \displaystyle 27 x^{2} \sqrt{y} - 5 e^{y} \sin{\left(x \right)}=-5 " /> </p> </p>
HTML for Canvas
<p> <p>Taking the derivative of both sides using implicit differentiation gives:
 <img class="equation_image" title="  \frac{27 x^{2} y'}{2 \sqrt{y}} + 54 x \sqrt{y} - 5 y' e^{y} \sin{\left(x \right)} - 5 e^{y} \cos{\left(x \right)} = 0  " src="/equation_images/%20%20%5Cfrac%7B27%20x%5E%7B2%7D%20y%27%7D%7B2%20%5Csqrt%7By%7D%7D%20%2B%2054%20x%20%5Csqrt%7By%7D%20-%205%20y%27%20e%5E%7By%7D%20%5Csin%7B%5Cleft%28x%20%5Cright%29%7D%20-%205%20e%5E%7By%7D%20%5Ccos%7B%5Cleft%28x%20%5Cright%29%7D%20%3D%200%20%20" alt="LaTeX:   \frac{27 x^{2} y'}{2 \sqrt{y}} + 54 x \sqrt{y} - 5 y' e^{y} \sin{\left(x \right)} - 5 e^{y} \cos{\left(x \right)} = 0  " data-equation-content="  \frac{27 x^{2} y'}{2 \sqrt{y}} + 54 x \sqrt{y} - 5 y' e^{y} \sin{\left(x \right)} - 5 e^{y} \cos{\left(x \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{2 \left(- 54 x y + 5 \sqrt{y} e^{y} \cos{\left(x \right)}\right)}{27 x^{2} - 10 \sqrt{y} e^{y} \sin{\left(x \right)}} " src="/equation_images/%20%5Cdisplaystyle%20y%27%20%3D%20%5Cfrac%7B2%20%5Cleft%28-%2054%20x%20y%20%2B%205%20%5Csqrt%7By%7D%20e%5E%7By%7D%20%5Ccos%7B%5Cleft%28x%20%5Cright%29%7D%5Cright%29%7D%7B27%20x%5E%7B2%7D%20-%2010%20%5Csqrt%7By%7D%20e%5E%7By%7D%20%5Csin%7B%5Cleft%28x%20%5Cright%29%7D%7D%20" alt="LaTeX:  \displaystyle y' = \frac{2 \left(- 54 x y + 5 \sqrt{y} e^{y} \cos{\left(x \right)}\right)}{27 x^{2} - 10 \sqrt{y} e^{y} \sin{\left(x \right)}} " data-equation-content=" \displaystyle y' = \frac{2 \left(- 54 x y + 5 \sqrt{y} e^{y} \cos{\left(x \right)}\right)}{27 x^{2} - 10 \sqrt{y} e^{y} \sin{\left(x \right)}} " /> </p> </p>