Use implicit differentiation where LaTeX: \displaystyle y is a function of LaTeX: \displaystyle x to find the derivative of LaTeX: \displaystyle - 2 x^{2} y^{2} - 7 \log{\left(x \right)} \log{\left(y \right)}=-23

Taking the derivative of both sides using implicit differentiation gives: LaTeX: - 4 x^{2} y y' - 4 x y^{2} - \frac{7 y' \log{\left(x \right)}}{y} - \frac{7 \log{\left(y \right)}}{x} = 0 Solving for LaTeX: \displaystyle y' gives LaTeX: \displaystyle y' = - \frac{y \left(4 x^{2} y^{2} + 7 \log{\left(y \right)}\right)}{x \left(4 x^{2} y^{2} + 7 \log{\left(x \right)}\right)}