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^{2} \right)} - 6 \sin{\left(x^{2} \right)} \sin{\left(y^{2} \right)}=6

Taking the derivative of both sides using implicit differentiation gives: LaTeX: 18 \sqrt{x} y y' \sin{\left(y^{2} \right)} - 12 x \sin{\left(y^{2} \right)} \cos{\left(x^{2} \right)} - 12 y y' \sin{\left(x^{2} \right)} \cos{\left(y^{2} \right)} - \frac{9 \cos{\left(y^{2} \right)}}{2 \sqrt{x}} = 0 Solving for LaTeX: \displaystyle y' gives LaTeX: \displaystyle y' = \frac{8 x^{\frac{3}{2}} \sin{\left(y^{2} \right)} \cos{\left(x^{2} \right)} + 3 \cos{\left(y^{2} \right)}}{4 y \left(- 2 \sqrt{x} \sin{\left(x^{2} \right)} \cos{\left(y^{2} \right)} + 3 x \sin{\left(y^{2} \right)}\right)}