A differentiable structure is a mathematical framework that allows us to define and analyze smooth functions on a set, typically a manifold. It provides a way to understand how to differentiate functions, ensuring that the necessary properties for calculus can be applied. This structure is essential in fields like differential geometry and theoretical physics. In a differentiable structure, we use charts and atlases to describe the manifold's local properties. Each chart provides a coordinate system, while the atlas is a collection of charts that cover the entire manifold. This organization helps mathematicians study complex shapes and spaces in a rigorous way.