Use implicit differentiation where LaTeX: \displaystyle y is a function of LaTeX: \displaystyle x to find the derivative of LaTeX: \displaystyle 6 \sqrt{x} y - 6 \log{\left(x \right)} \log{\left(y \right)}=13

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