Use implicit differentiation where LaTeX: \displaystyle y is a function of LaTeX: \displaystyle x to find the derivative of LaTeX: \displaystyle - 9 e^{y} \cos{\left(x^{3} \right)} + 6 \log{\left(y \right)} \sin{\left(x^{3} \right)}=-35

Taking the derivative of both sides using implicit differentiation gives: LaTeX: 27 x^{2} e^{y} \sin{\left(x^{3} \right)} + 18 x^{2} \log{\left(y \right)} \cos{\left(x^{3} \right)} - 9 y' e^{y} \cos{\left(x^{3} \right)} + \frac{6 y' \sin{\left(x^{3} \right)}}{y} = 0 Solving for LaTeX: \displaystyle y' gives LaTeX: \displaystyle y' = \frac{3 x^{2} y \left(3 e^{y} \sin{\left(x^{3} \right)} + 2 \log{\left(y \right)} \cos{\left(x^{3} \right)}\right)}{3 y e^{y} \cos{\left(x^{3} \right)} - 2 \sin{\left(x^{3} \right)}}