The definition of a function being differentiable at a number and the derivative as a number usually appears in a calculus text after motivation connected to interpretations of the derivative as:
1. the slope of the tangent line at a point on a curve that is the graph of a function,
2. the instantaneous velocity at a given instant of an object moving on a straight line,  and
3. the instantaneous rate of change of a dependent variable with respect to an independent variable at a given value.
Approximations and limit concepts tie these interpretations of the derivative to related measures-
1. the slope of secant lines between points on a curve that is the graph of a function,
2. average velocities of an object moving on a straight line for time intervals, and
3. average rates of change of a dependent variable with respect to an independent variable for intervals of values.
The definition of the derivative as a function usually comes after the number definition leading to the lengthy evolution of the calculus for derivatives.

Definitions for one sided derivatives usually appear directly after the number (point) definitions to allow further consideration of functions defined with domains other than open intervals.

Here are traditional versions of these definitions connected to visualizations with mapping diagrams.

CCD.DDN: Definition of Derivative as a Number (Differentiable at a Point):

CCD.DDN4S: Using the Definition of Derivative as a Number with Four Steps:

CCD.DDF: Definition of Derivative as a Function (Differentiable on a Domain):

CCD.DDOS: Definition of a One Sided Derivative:

Dynamic visualization of the definitions of the derivative:

Examples of Derivatives and Singularities (Non-differentiability) with Mapping Diagrams and Graphs.
The connection between limits and the derivative found with singularities are reinforced with key examples of continuous functions with isolated point singularities. These examples are improved by using mapping diagrams along with graphs.

CCD.DDS.AV. Example 1.  Singularities with the absolute value function.

CCD.DDS.R. Example.2.  Singularities with a root function.