Use implicit differentiation where $LaTeX: \displaystyle y$ is a function of $LaTeX: \displaystyle x$ to find the derivative of $LaTeX: \displaystyle 40 \sqrt{x} \sqrt{y} - 4 y \log{\left(x \right)}=14$

Taking the derivative of both sides using implicit differentiation gives: $LaTeX: \frac{20 \sqrt{x} y'}{\sqrt{y}} - 4 y' \log{\left(x \right)} - \frac{4 y}{x} + \frac{20 \sqrt{y}}{\sqrt{x}} = 0$ Solving for $LaTeX: \displaystyle y'$ gives $LaTeX: \displaystyle y' = \frac{- \sqrt{x} y^{\frac{3}{2}} + 5 x y}{x^{\frac{3}{2}} \sqrt{y} \log{\left(x \right)} - 5 x^{2}}$