what does differentiable mean in calculus

In calculus, a function is considered differentiable if its derivative exists at each point in its domain. This means that the slope of the tangent line equals the limit of the function at a given point. In other words, a function is differentiable if it has a non-vertical tangent line at each point in its domain. To determine differentiability, one can use limits and continuity. Specifically, a function is differentiable on an open interval (a,b) if the limit of the difference quotient exists for every c in (a,b) . It is important to note that a differentiable function is always continuous, but the converse is not necessarily true. A continuous function is not necessarily differentiable, but a differentiable function is necessarily continuous at every point where it is differentiable.