Use implicit differentiation where $LaTeX: \displaystyle y$ is a function of $LaTeX: \displaystyle x$ to find the derivative of $LaTeX: \displaystyle \log{\left(x \right)} - 3 \sin{\left(y \right)} \cos{\left(x \right)}=-14$

Taking the derivative of both sides using implicit differentiation gives: $LaTeX: - 3 y' \cos{\left(x \right)} \cos{\left(y \right)} + 3 \sin{\left(x \right)} \sin{\left(y \right)} + \frac{1}{x} = 0$ Solving for $LaTeX: \displaystyle y'$ gives $LaTeX: \displaystyle y' = \tan{\left(x \right)} \tan{\left(y \right)} + \frac{1}{3 x \cos{\left(x \right)} \cos{\left(y \right)}}$