Use implicit differentiation where $LaTeX: \displaystyle y$ is a function of $LaTeX: \displaystyle x$ to find the derivative of $LaTeX: \displaystyle - 9 y \cos{\left(x \right)} + e^{x^{3}} e^{y^{3}}=20$

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