Use implicit differentiation where LaTeX: \displaystyle y is a function of LaTeX: \displaystyle x to find the derivative of LaTeX: \displaystyle 2 x^{3} \sin{\left(y \right)} - 3 \cos{\left(x^{2} \right)} \cos{\left(y^{2} \right)}=19

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