Use implicit differentiation where LaTeX: \displaystyle y is a function of LaTeX: \displaystyle x to find the derivative of LaTeX: \displaystyle 7 e^{x} \cos{\left(y^{3} \right)} + 8 \log{\left(y \right)} \sin{\left(x^{2} \right)}=-30

Taking the derivative of both sides using implicit differentiation gives: LaTeX: 16 x \log{\left(y \right)} \cos{\left(x^{2} \right)} - 21 y^{2} y' e^{x} \sin{\left(y^{3} \right)} + 7 e^{x} \cos{\left(y^{3} \right)} + \frac{8 y' \sin{\left(x^{2} \right)}}{y} = 0 Solving for LaTeX: \displaystyle y' gives LaTeX: \displaystyle y' = \frac{y \left(16 x \log{\left(y \right)} \cos{\left(x^{2} \right)} + 7 e^{x} \cos{\left(y^{3} \right)}\right)}{21 y^{3} e^{x} \sin{\left(y^{3} \right)} - 8 \sin{\left(x^{2} \right)}}