Use implicit differentiation where LaTeX: \displaystyle y is a function of LaTeX: \displaystyle x to find the derivative of LaTeX: \displaystyle - \sqrt{6} \sqrt{x} \sin{\left(y^{2} \right)} - 3 e^{y^{2}} \log{\left(x \right)}=39

Taking the derivative of both sides using implicit differentiation gives: LaTeX: - 2 \sqrt{6} \sqrt{x} y y' \cos{\left(y^{2} \right)} - 6 y y' e^{y^{2}} \log{\left(x \right)} - \frac{3 e^{y^{2}}}{x} - \frac{\sqrt{6} \sin{\left(y^{2} \right)}}{2 \sqrt{x}} = 0 Solving for LaTeX: \displaystyle y' gives LaTeX: \displaystyle y' = - \frac{6 \sqrt{x} e^{y^{2}} + \sqrt{6} x \sin{\left(y^{2} \right)}}{4 y \left(3 x^{\frac{3}{2}} e^{y^{2}} \log{\left(x \right)} + \sqrt{6} x^{2} \cos{\left(y^{2} \right)}\right)}