Use implicit differentiation where LaTeX: \displaystyle y is a function of LaTeX: \displaystyle x to find the derivative of LaTeX: \displaystyle - 9 x^{3} \sin{\left(y^{3} \right)} - 5 \log{\left(y \right)} \cos{\left(x^{3} \right)}=21

Taking the derivative of both sides using implicit differentiation gives: LaTeX: - 27 x^{3} y^{2} y' \cos{\left(y^{3} \right)} + 15 x^{2} \log{\left(y \right)} \sin{\left(x^{3} \right)} - 27 x^{2} \sin{\left(y^{3} \right)} - \frac{5 y' \cos{\left(x^{3} \right)}}{y} = 0 Solving for LaTeX: \displaystyle y' gives LaTeX: \displaystyle y' = \frac{3 x^{2} y \left(5 \log{\left(y \right)} \sin{\left(x^{3} \right)} - 9 \sin{\left(y^{3} \right)}\right)}{27 x^{3} y^{3} \cos{\left(y^{3} \right)} + 5 \cos{\left(x^{3} \right)}}