what does differentiable mean in calculus

In calculus, differentiable means a function has a derivative at a point or over an interval, so its slope is well-defined there. In plain language, the graph is smooth enough at that spot that you can draw a tangent line without hitting a sharp corner, cusp, or break.

Quick idea

• If a function is differentiable at x=ax=ax=a, then the limit of the slope of the secant lines exists as the input approaches aaa.

• A differentiable function is always continuous at that point, but a continuous function is not always differentiable.

Simple example

• f(x)=x2f(x)=x^2f(x)=x2 is differentiable everywhere, because its slope changes smoothly.

• f(x)=∣x∣f(x)=|x|f(x)=∣x∣ is continuous, but it is not differentiable at x=0x=0x=0 because the graph has a sharp corner there

One-sentence version

Differentiable means “has a derivative,” which usually means the graph is smooth enough to have a tangent line at that point.”

TL;DR: differentiable = derivative exists; continuous = no breaks; differentiable is stronger than continuous.