Explore partial derivatives and multivariable optimization through interactive SageMath computations including critical point analysis, Lagrange multipliers, and constrained optimization problems. This hands-on Jupyter notebook covers second derivative tests, Hessian matrices, saddle point identification, and practical optimization applications in economics and engineering. CoCalc provides pre-configured computational tools for symbolic differentiation, 3D surface plotting, and gradient descent visualization, allowing students to solve complex optimization problems and understand multivariable calculus concepts through immediate computational feedback.