Use implicit differentiation where LaTeX: \displaystyle y is a function of LaTeX: \displaystyle x to find the derivative of LaTeX: \displaystyle 4 y \log{\left(x \right)} + 7 e^{y^{3}} \cos{\left(x^{3} \right)}=-45

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