Use implicit differentiation where LaTeX: \displaystyle y is a function of LaTeX: \displaystyle x to find the derivative of LaTeX: \displaystyle - 7 \sqrt{7} \sqrt{x} \sin{\left(y^{2} \right)} - 6 y \cos{\left(x^{3} \right)}=-37

Taking the derivative of both sides using implicit differentiation gives: LaTeX: - 14 \sqrt{7} \sqrt{x} y y' \cos{\left(y^{2} \right)} + 18 x^{2} y \sin{\left(x^{3} \right)} - 6 y' \cos{\left(x^{3} \right)} - \frac{7 \sqrt{7} \sin{\left(y^{2} \right)}}{2 \sqrt{x}} = 0 Solving for LaTeX: \displaystyle y' gives LaTeX: \displaystyle y' = \frac{36 x^{\frac{5}{2}} y \sin{\left(x^{3} \right)} - 7 \sqrt{7} \sin{\left(y^{2} \right)}}{4 \left(3 \sqrt{x} \cos{\left(x^{3} \right)} + 7 \sqrt{7} x y \cos{\left(y^{2} \right)}\right)}