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

Taking the derivative of both sides using implicit differentiation gives: LaTeX: 12 x^{2} e^{y^{2}} \cos{\left(x^{3} \right)} - 4 x y' \sin{\left(y \right)} + 8 y y' e^{y^{2}} \sin{\left(x^{3} \right)} + 4 \cos{\left(y \right)} = 0 Solving for LaTeX: \displaystyle y' gives LaTeX: \displaystyle y' = \frac{3 x^{2} e^{y^{2}} \cos{\left(x^{3} \right)} + \cos{\left(y \right)}}{x \sin{\left(y \right)} - 2 y e^{y^{2}} \sin{\left(x^{3} \right)}}