Use implicit differentiation where LaTeX: \displaystyle y is a function of LaTeX: \displaystyle x to find the derivative of LaTeX: \displaystyle 9 x^{3} y^{3} - 4 \log{\left(x \right)} \cos{\left(y \right)}=7

Taking the derivative of both sides using implicit differentiation gives: LaTeX: 27 x^{3} y^{2} y' + 27 x^{2} y^{3} + 4 y' \log{\left(x \right)} \sin{\left(y \right)} - \frac{4 \cos{\left(y \right)}}{x} = 0 Solving for LaTeX: \displaystyle y' gives LaTeX: \displaystyle y' = \frac{- 27 x^{3} y^{3} + 4 \cos{\left(y \right)}}{x \left(27 x^{3} y^{2} + 4 \log{\left(x \right)} \sin{\left(y \right)}\right)}