Use implicit differentiation where LaTeX: \displaystyle y is a function of LaTeX: \displaystyle x to find the derivative of LaTeX: \displaystyle - 6 x^{3} \log{\left(y \right)} + 7 e^{y^{2}} \sin{\left(x \right)}=3

Taking the derivative of both sides using implicit differentiation gives: LaTeX: - \frac{6 x^{3} y'}{y} - 18 x^{2} \log{\left(y \right)} + 14 y y' e^{y^{2}} \sin{\left(x \right)} + 7 e^{y^{2}} \cos{\left(x \right)} = 0 Solving for LaTeX: \displaystyle y' gives LaTeX: \displaystyle y' = \frac{y \left(- 18 x^{2} \log{\left(y \right)} + 7 e^{y^{2}} \cos{\left(x \right)}\right)}{2 \left(3 x^{3} - 7 y^{2} e^{y^{2}} \sin{\left(x \right)}\right)}