Use implicit differentiation where LaTeX: \displaystyle y is a function of LaTeX: \displaystyle x to find the derivative of LaTeX: \displaystyle - 56 \sqrt{x} \sqrt{y} + 9 \log{\left(x \right)} \log{\left(y \right)}=21

Taking the derivative of both sides using implicit differentiation gives: LaTeX: - \frac{28 \sqrt{x} y'}{\sqrt{y}} + \frac{9 y' \log{\left(x \right)}}{y} + \frac{9 \log{\left(y \right)}}{x} - \frac{28 \sqrt{y}}{\sqrt{x}} = 0 Solving for LaTeX: \displaystyle y' gives LaTeX: \displaystyle y' = \frac{- 9 \sqrt{x} y^{\frac{3}{2}} \log{\left(y \right)} + 28 x y^{2}}{9 x^{\frac{3}{2}} \sqrt{y} \log{\left(x \right)} - 28 x^{2} y}