A differentiable function is a type of mathematical function that has a derivative at every point in its domain. This means that the function's graph has a well-defined tangent line at each point, indicating that it is smooth and continuous without any sharp corners or breaks. In calculus, the concept of differentiability is crucial because it allows us to analyze how functions change. If a function is differentiable, we can use tools like the derivative to find rates of change, slopes, and optimize values, making it an essential concept in both pure and applied mathematics.