what does it mean for a function to be differentiable

A function is differentiable at a point if its derivative exists at that point, meaning it has a well-defined tangent line there. More generally, a function is differentiable on an interval if it is differentiable at every point in that interval. Differentiability means the function is smooth at the point or interval—there are no sharp corners, cusps, or vertical tangents. Also, differentiability implies continuity, but continuity alone does not guarantee differentiability. In more formal terms, for a function fff defined on a domain, the derivative f′(a)f'(a)f′(a) at a point aaa exists if the limit

f′(a)=lim⁡h→0f(a+h)−f(a)hf'(a)=\lim_{h\to 0}\frac{f(a+h)-f(a)}{h}f′(a)=h→0lim​hf(a+h)−f(a)​

exists. If this limit exists, the function is differentiable at aaa and is locally well approximated by a linear function (its tangent line) near aaa.