Use implicit differentiation where LaTeX: \displaystyle y is a function of LaTeX: \displaystyle x to find the derivative of LaTeX: \displaystyle - 18 \sqrt{x} \cos{\left(y^{2} \right)} - 8 e^{y} \cos{\left(x^{2} \right)}=15

Taking the derivative of both sides using implicit differentiation gives: LaTeX: 36 \sqrt{x} y y' \sin{\left(y^{2} \right)} + 16 x e^{y} \sin{\left(x^{2} \right)} - 8 y' e^{y} \cos{\left(x^{2} \right)} - \frac{9 \cos{\left(y^{2} \right)}}{\sqrt{x}} = 0 Solving for LaTeX: \displaystyle y' gives LaTeX: \displaystyle y' = \frac{16 x^{\frac{3}{2}} e^{y} \sin{\left(x^{2} \right)} - 9 \cos{\left(y^{2} \right)}}{4 \left(2 \sqrt{x} e^{y} \cos{\left(x^{2} \right)} - 9 x y \sin{\left(y^{2} \right)}\right)}