Use implicit differentiation where $LaTeX: \displaystyle y$ is a function of $LaTeX: \displaystyle x$ to find the derivative of $LaTeX: \displaystyle - 9 \sqrt{6} \sqrt{x} \log{\left(y \right)} - \log{\left(x \right)} \sin{\left(y \right)}=49$

Taking the derivative of both sides using implicit differentiation gives: $LaTeX: - \frac{9 \sqrt{6} \sqrt{x} y'}{y} - y' \log{\left(x \right)} \cos{\left(y \right)} - \frac{\sin{\left(y \right)}}{x} - \frac{9 \sqrt{6} \log{\left(y \right)}}{2 \sqrt{x}} = 0$ Solving for $LaTeX: \displaystyle y'$ gives $LaTeX: \displaystyle y' = - \frac{y \left(2 \sqrt{x} \sin{\left(y \right)} + 9 \sqrt{6} x \log{\left(y \right)}\right)}{2 x^{\frac{3}{2}} y \log{\left(x \right)} \cos{\left(y \right)} + 18 \sqrt{6} x^{2}}$