Use implicit differentiation where LaTeX: \displaystyle y is a function of LaTeX: \displaystyle x to find the derivative of LaTeX: \displaystyle 12 \sqrt{x} \cos{\left(y \right)} - y^{3} \log{\left(x \right)}=-21

Taking the derivative of both sides using implicit differentiation gives: LaTeX: - 12 \sqrt{x} y' \sin{\left(y \right)} - 3 y^{2} y' \log{\left(x \right)} - \frac{y^{3}}{x} + \frac{6 \cos{\left(y \right)}}{\sqrt{x}} = 0 Solving for LaTeX: \displaystyle y' gives LaTeX: \displaystyle y' = \frac{- \sqrt{x} y^{3} + 6 x \cos{\left(y \right)}}{3 \left(x^{\frac{3}{2}} y^{2} \log{\left(x \right)} + 4 x^{2} \sin{\left(y \right)}\right)}