Use implicit differentiation where LaTeX: \displaystyle y is a function of LaTeX: \displaystyle x to find the derivative of LaTeX: \displaystyle 12 \sqrt{x} \sqrt{y} - 5 \log{\left(y \right)} \sin{\left(x \right)}=-5

Taking the derivative of both sides using implicit differentiation gives: LaTeX: \frac{6 \sqrt{x} y'}{\sqrt{y}} - 5 \log{\left(y \right)} \cos{\left(x \right)} - \frac{5 y' \sin{\left(x \right)}}{y} + \frac{6 \sqrt{y}}{\sqrt{x}} = 0 Solving for LaTeX: \displaystyle y' gives LaTeX: \displaystyle y' = \frac{- 5 \sqrt{x} y^{\frac{3}{2}} \log{\left(y \right)} \cos{\left(x \right)} + 6 y^{2}}{5 \sqrt{x} \sqrt{y} \sin{\left(x \right)} - 6 x y}