Use implicit differentiation where $LaTeX: \displaystyle y$ is a function of $LaTeX: \displaystyle x$ to find the derivative of $LaTeX: \displaystyle - 9 e^{x} \sin{\left(y \right)} + \log{\left(x \right)}=-23$

Taking the derivative of both sides using implicit differentiation gives: $LaTeX: - 9 y' e^{x} \cos{\left(y \right)} - 9 e^{x} \sin{\left(y \right)} + \frac{1}{x} = 0$ Solving for $LaTeX: \displaystyle y'$ gives $LaTeX: \displaystyle y' = - \tan{\left(y \right)} + \frac{e^{- x}}{9 x \cos{\left(y \right)}}$