Use implicit differentiation where LaTeX: \displaystyle y is a function of LaTeX: \displaystyle x to find the derivative of LaTeX: \displaystyle 21 \sqrt{y} \sin{\left(x^{3} \right)} + 8 \log{\left(x \right)} \log{\left(y \right)}=-27

Taking the derivative of both sides using implicit differentiation gives: LaTeX: 63 x^{2} \sqrt{y} \cos{\left(x^{3} \right)} + \frac{8 y' \log{\left(x \right)}}{y} + \frac{21 y' \sin{\left(x^{3} \right)}}{2 \sqrt{y}} + \frac{8 \log{\left(y \right)}}{x} = 0 Solving for LaTeX: \displaystyle y' gives LaTeX: \displaystyle y' = - \frac{126 x^{3} y^{2} \cos{\left(x^{3} \right)} + 16 y^{\frac{3}{2}} \log{\left(y \right)}}{x \left(16 \sqrt{y} \log{\left(x \right)} + 21 y \sin{\left(x^{3} \right)}\right)}