Use implicit differentiation where LaTeX: \displaystyle y is a function of LaTeX: \displaystyle x to find the derivative of LaTeX: \displaystyle - 3 \sqrt{6} \sqrt{x} \sin{\left(y^{2} \right)} + 8 \sqrt{6} \sqrt{y} \log{\left(x \right)}=-35

Taking the derivative of both sides using implicit differentiation gives: LaTeX: - 6 \sqrt{6} \sqrt{x} y y' \cos{\left(y^{2} \right)} + \frac{4 \sqrt{6} y' \log{\left(x \right)}}{\sqrt{y}} + \frac{8 \sqrt{6} \sqrt{y}}{x} - \frac{3 \sqrt{6} \sin{\left(y^{2} \right)}}{2 \sqrt{x}} = 0 Solving for LaTeX: \displaystyle y' gives LaTeX: \displaystyle y' = \frac{- 16 \sqrt{x} y + 3 x \sqrt{y} \sin{\left(y^{2} \right)}}{4 \left(2 x^{\frac{3}{2}} \log{\left(x \right)} - 3 x^{2} y^{\frac{3}{2}} \cos{\left(y^{2} \right)}\right)}