Use implicit differentiation where LaTeX: \displaystyle y is a function of LaTeX: \displaystyle x to find the derivative of LaTeX: \displaystyle 9 y \sin{\left(x^{2} \right)} + 4 e^{y^{2}} \log{\left(x \right)}=-21

Taking the derivative of both sides using implicit differentiation gives: LaTeX: 18 x y \cos{\left(x^{2} \right)} + 8 y y' e^{y^{2}} \log{\left(x \right)} + 9 y' \sin{\left(x^{2} \right)} + \frac{4 e^{y^{2}}}{x} = 0 Solving for LaTeX: \displaystyle y' gives LaTeX: \displaystyle y' = - \frac{18 x^{2} y \cos{\left(x^{2} \right)} + 4 e^{y^{2}}}{x \left(8 y e^{y^{2}} \log{\left(x \right)} + 9 \sin{\left(x^{2} \right)}\right)}