Use implicit differentiation where LaTeX: \displaystyle y is a function of LaTeX: \displaystyle x to find the derivative of LaTeX: \displaystyle - x \cos{\left(y \right)} - 6 \log{\left(y \right)} \sin{\left(x \right)}=-30

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