Use implicit differentiation where LaTeX: \displaystyle y is a function of LaTeX: \displaystyle x to find the derivative of LaTeX: \displaystyle - 6 e^{x^{3}} \cos{\left(y \right)} - 7 \log{\left(x \right)} \sin{\left(y^{3} \right)}=-49

Taking the derivative of both sides using implicit differentiation gives: LaTeX: - 18 x^{2} e^{x^{3}} \cos{\left(y \right)} - 21 y^{2} y' \log{\left(x \right)} \cos{\left(y^{3} \right)} + 6 y' e^{x^{3}} \sin{\left(y \right)} - \frac{7 \sin{\left(y^{3} \right)}}{x} = 0 Solving for LaTeX: \displaystyle y' gives LaTeX: \displaystyle y' = - \frac{18 x^{3} e^{x^{3}} \cos{\left(y \right)} + 7 \sin{\left(y^{3} \right)}}{3 x \left(7 y^{2} \log{\left(x \right)} \cos{\left(y^{3} \right)} - 2 e^{x^{3}} \sin{\left(y \right)}\right)}