Use implicit differentiation where LaTeX: \displaystyle y is a function of LaTeX: \displaystyle x to find the derivative of LaTeX: \displaystyle - 7 y^{2} \sin{\left(x^{3} \right)} - 4 \log{\left(x \right)} \cos{\left(y^{3} \right)}=-4

Taking the derivative of both sides using implicit differentiation gives: LaTeX: - 21 x^{2} y^{2} \cos{\left(x^{3} \right)} + 12 y^{2} y' \log{\left(x \right)} \sin{\left(y^{3} \right)} - 14 y y' \sin{\left(x^{3} \right)} - \frac{4 \cos{\left(y^{3} \right)}}{x} = 0 Solving for LaTeX: \displaystyle y' gives LaTeX: \displaystyle y' = \frac{21 x^{3} y^{2} \cos{\left(x^{3} \right)} + 4 \cos{\left(y^{3} \right)}}{2 x y \left(6 y \log{\left(x \right)} \sin{\left(y^{3} \right)} - 7 \sin{\left(x^{3} \right)}\right)}