Use implicit differentiation where LaTeX: \displaystyle y is a function of LaTeX: \displaystyle x to find the derivative of LaTeX: \displaystyle - 2 \sqrt{3} \sqrt{x} y^{3} + 6 \log{\left(y \right)} \sin{\left(x \right)}=21

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