Use implicit differentiation where $LaTeX: \displaystyle y$ is a function of $LaTeX: \displaystyle x$ to find the derivative of $LaTeX: \displaystyle 9 \sqrt{x} \cos{\left(y \right)} - 9 \sqrt{y} e^{x}=45$

Taking the derivative of both sides using implicit differentiation gives: $LaTeX: - 9 \sqrt{x} y' \sin{\left(y \right)} - 9 \sqrt{y} e^{x} - \frac{9 y' e^{x}}{2 \sqrt{y}} + \frac{9 \cos{\left(y \right)}}{2 \sqrt{x}} = 0$ Solving for $LaTeX: \displaystyle y'$ gives $LaTeX: \displaystyle y' = \frac{- 2 \sqrt{x} y e^{x} + \sqrt{y} \cos{\left(y \right)}}{\sqrt{x} e^{x} + 2 x \sqrt{y} \sin{\left(y \right)}}$