Use implicit differentiation where LaTeX: \displaystyle y is a function of LaTeX: \displaystyle x to find the derivative of LaTeX: \displaystyle - 2 x^{2} \sin{\left(y^{2} \right)} - \sin{\left(x^{2} \right)} \cos{\left(y^{3} \right)}=-34

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