Use implicit differentiation where LaTeX: \displaystyle y is a function of LaTeX: \displaystyle x to find the derivative of LaTeX: \displaystyle 9 y^{2} \cos{\left(x^{3} \right)} + 2 \sin{\left(x^{2} \right)} \cos{\left(y^{3} \right)}=-43

Taking the derivative of both sides using implicit differentiation gives: LaTeX: - 27 x^{2} y^{2} \sin{\left(x^{3} \right)} + 4 x \cos{\left(x^{2} \right)} \cos{\left(y^{3} \right)} - 6 y^{2} y' \sin{\left(x^{2} \right)} \sin{\left(y^{3} \right)} + 18 y y' \cos{\left(x^{3} \right)} = 0 Solving for LaTeX: \displaystyle y' gives LaTeX: \displaystyle y' = \frac{x \left(- 27 x y^{2} \sin{\left(x^{3} \right)} + 4 \cos{\left(x^{2} \right)} \cos{\left(y^{3} \right)}\right)}{6 y \left(y \sin{\left(x^{2} \right)} \sin{\left(y^{3} \right)} - 3 \cos{\left(x^{3} \right)}\right)}