Use implicit differentiation where LaTeX: \displaystyle y is a function of LaTeX: \displaystyle x to find the derivative of LaTeX: \displaystyle - 5 y^{3} \sin{\left(x \right)} - 4 e^{x^{3}} \cos{\left(y^{2} \right)}=22

Taking the derivative of both sides using implicit differentiation gives: LaTeX: - 12 x^{2} e^{x^{3}} \cos{\left(y^{2} \right)} - 5 y^{3} \cos{\left(x \right)} - 15 y^{2} y' \sin{\left(x \right)} + 8 y y' e^{x^{3}} \sin{\left(y^{2} \right)} = 0 Solving for LaTeX: \displaystyle y' gives LaTeX: \displaystyle y' = \frac{12 x^{2} e^{x^{3}} \cos{\left(y^{2} \right)} + 5 y^{3} \cos{\left(x \right)}}{y \left(- 15 y \sin{\left(x \right)} + 8 e^{x^{3}} \sin{\left(y^{2} \right)}\right)}