Use implicit differentiation where LaTeX: \displaystyle y is a function of LaTeX: \displaystyle x to find the derivative of LaTeX: \displaystyle - 6 x \sqrt{y} - 6 \log{\left(x \right)} \log{\left(y \right)}=46

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