Use implicit differentiation where LaTeX: \displaystyle y is a function of LaTeX: \displaystyle x to find the derivative of LaTeX: \displaystyle - 6 \sqrt{6} \sqrt{y} \cos{\left(x^{2} \right)} + 9 \log{\left(x \right)} \log{\left(y \right)}=4

Taking the derivative of both sides using implicit differentiation gives: LaTeX: 12 \sqrt{6} x \sqrt{y} \sin{\left(x^{2} \right)} + \frac{9 y' \log{\left(x \right)}}{y} - \frac{3 \sqrt{6} y' \cos{\left(x^{2} \right)}}{\sqrt{y}} + \frac{9 \log{\left(y \right)}}{x} = 0 Solving for LaTeX: \displaystyle y' gives LaTeX: \displaystyle y' = \frac{4 \sqrt{6} x^{2} y^{2} \sin{\left(x^{2} \right)} + 3 y^{\frac{3}{2}} \log{\left(y \right)}}{x \left(- 3 \sqrt{y} \log{\left(x \right)} + \sqrt{6} y \cos{\left(x^{2} \right)}\right)}