Use implicit differentiation where $LaTeX: \displaystyle y$ is a function of $LaTeX: \displaystyle x$ to find the derivative of $LaTeX: \displaystyle 27 x^{2} \sqrt{y} - 5 e^{y} \sin{\left(x \right)}=-5$

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