Use implicit differentiation where $LaTeX: \displaystyle y$ is a function of $LaTeX: \displaystyle x$ to find the derivative of $LaTeX: \displaystyle - 6 y^{3} \cos{\left(x \right)} - 5 e^{x^{3}} \cos{\left(y \right)}=47$

Taking the derivative of both sides using implicit differentiation gives: $LaTeX: - 15 x^{2} e^{x^{3}} \cos{\left(y \right)} + 6 y^{3} \sin{\left(x \right)} - 18 y^{2} y' \cos{\left(x \right)} + 5 y' e^{x^{3}} \sin{\left(y \right)} = 0$ Solving for $LaTeX: \displaystyle y'$ gives $LaTeX: \displaystyle y' = \frac{3 \left(- 5 x^{2} e^{x^{3}} \cos{\left(y \right)} + 2 y^{3} \sin{\left(x \right)}\right)}{18 y^{2} \cos{\left(x \right)} - 5 e^{x^{3}} \sin{\left(y \right)}}$