Use implicit differentiation where LaTeX: \displaystyle y is a function of LaTeX: \displaystyle x to find the derivative of LaTeX: \displaystyle - 27 \sqrt{x} \sqrt{y} + 4 \sin{\left(x^{2} \right)} \sin{\left(y^{2} \right)}=-5

Taking the derivative of both sides using implicit differentiation gives: LaTeX: - \frac{27 \sqrt{x} y'}{2 \sqrt{y}} + 8 x \sin{\left(y^{2} \right)} \cos{\left(x^{2} \right)} + 8 y y' \sin{\left(x^{2} \right)} \cos{\left(y^{2} \right)} - \frac{27 \sqrt{y}}{2 \sqrt{x}} = 0 Solving for LaTeX: \displaystyle y' gives LaTeX: \displaystyle y' = \frac{- 16 x^{\frac{3}{2}} \sqrt{y} \sin{\left(y^{2} \right)} \cos{\left(x^{2} \right)} + 27 y}{16 \sqrt{x} y^{\frac{3}{2}} \sin{\left(x^{2} \right)} \cos{\left(y^{2} \right)} - 27 x}