Use implicit differentiation where LaTeX: \displaystyle y is a function of LaTeX: \displaystyle x to find the derivative of LaTeX: \displaystyle - 35 \sqrt{x} \sqrt{y} + 3 \sin{\left(x^{2} \right)} \sin{\left(y^{2} \right)}=37

Taking the derivative of both sides using implicit differentiation gives: LaTeX: - \frac{35 \sqrt{x} y'}{2 \sqrt{y}} + 6 x \sin{\left(y^{2} \right)} \cos{\left(x^{2} \right)} + 6 y y' \sin{\left(x^{2} \right)} \cos{\left(y^{2} \right)} - \frac{35 \sqrt{y}}{2 \sqrt{x}} = 0 Solving for LaTeX: \displaystyle y' gives LaTeX: \displaystyle y' = \frac{- 12 x^{\frac{3}{2}} \sqrt{y} \sin{\left(y^{2} \right)} \cos{\left(x^{2} \right)} + 35 y}{12 \sqrt{x} y^{\frac{3}{2}} \sin{\left(x^{2} \right)} \cos{\left(y^{2} \right)} - 35 x}