Differentiable vs. Nondifferentiable Functions
Key Questions

geometrically, the function
#f# is differentiable at#a# if it has a nonvertical tangent at the corresponding point on the graph, that is, at#(a,f(a))# . That means that the limit
#lim_{x\to a} (f(x)f(a))/(xa)# exists (i.e, is a finite number, which is the slope of this tangent line). When this limit exist, it is called derivative of#f# at#a# and denoted#f'(a)# or#(df)/dx (a)# . So a point where the function is not differentiable is a point where this limit does not exist, that is, is either infinite (case of a vertical tangent), where the function is discontinuous, or where there are two different onesided limits (a cusp, like for#f(x)=x# at 0). See definition of the derivative and derivative as a function.
Questions
Derivatives

Tangent Line to a Curve

Normal Line to a Tangent

Slope of a Curve at a Point

Average Velocity

Instantaneous Velocity

Limit Definition of Derivative

First Principles Example 1: x²

First Principles Example 2: x³

First Principles Example 3: square root of x

Standard Notation and Terminology

Differentiable vs. Nondifferentiable Functions

Rate of Change of a Function

Average Rate of Change Over an Interval

Instantaneous Rate of Change at a Point