Use implicit differentiation where LaTeX: \displaystyle y is a function of LaTeX: \displaystyle x to find the derivative of LaTeX: \displaystyle 27 \sqrt{x} \cos{\left(y \right)} - 12 \sqrt{y} e^{x}=-17

Taking the derivative of both sides using implicit differentiation gives: LaTeX: - 27 \sqrt{x} y' \sin{\left(y \right)} - 12 \sqrt{y} e^{x} - \frac{6 y' e^{x}}{\sqrt{y}} + \frac{27 \cos{\left(y \right)}}{2 \sqrt{x}} = 0 Solving for LaTeX: \displaystyle y' gives LaTeX: \displaystyle y' = \frac{- 8 \sqrt{x} y e^{x} + 9 \sqrt{y} \cos{\left(y \right)}}{2 \left(2 \sqrt{x} e^{x} + 9 x \sqrt{y} \sin{\left(y \right)}\right)}