Use implicit differentiation where LaTeX: \displaystyle y is a function of LaTeX: \displaystyle x to find the derivative of LaTeX: \displaystyle 9 \sqrt{3} \sqrt{y} \log{\left(x \right)} - 2 \sin{\left(y^{3} \right)} \cos{\left(x \right)}=-47

Taking the derivative of both sides using implicit differentiation gives: LaTeX: - 6 y^{2} y' \cos{\left(x \right)} \cos{\left(y^{3} \right)} + 2 \sin{\left(x \right)} \sin{\left(y^{3} \right)} + \frac{9 \sqrt{3} y' \log{\left(x \right)}}{2 \sqrt{y}} + \frac{9 \sqrt{3} \sqrt{y}}{x} = 0 Solving for LaTeX: \displaystyle y' gives LaTeX: \displaystyle y' = \frac{2 \left(2 x \sqrt{y} \sin{\left(x \right)} \sin{\left(y^{3} \right)} + 9 \sqrt{3} y\right)}{3 x \left(4 y^{\frac{5}{2}} \cos{\left(x \right)} \cos{\left(y^{3} \right)} - 3 \sqrt{3} \log{\left(x \right)}\right)}