Use implicit differentiation where $LaTeX: \displaystyle y$ is a function of $LaTeX: \displaystyle x$ to find the derivative of $LaTeX: \displaystyle 16 \sqrt{x} e^{y} + 10 \sqrt{y} \cos{\left(x \right)}=27$

Taking the derivative of both sides using implicit differentiation gives: $LaTeX: 16 \sqrt{x} y' e^{y} - 10 \sqrt{y} \sin{\left(x \right)} + \frac{5 y' \cos{\left(x \right)}}{\sqrt{y}} + \frac{8 e^{y}}{\sqrt{x}} = 0$ Solving for $LaTeX: \displaystyle y'$ gives $LaTeX: \displaystyle y' = \frac{2 \left(5 \sqrt{x} y \sin{\left(x \right)} - 4 \sqrt{y} e^{y}\right)}{5 \sqrt{x} \cos{\left(x \right)} + 16 x \sqrt{y} e^{y}}$