Use implicit differentiation where LaTeX: \displaystyle y is a function of LaTeX: \displaystyle x to find the derivative of LaTeX: \displaystyle - 7 e^{y^{3}} \log{\left(x \right)} - \sin{\left(x^{2} \right)} \cos{\left(y \right)}=-1

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