WHAT IS THE MEANING OF DIFFERENTIATION

Differentiation

Introduction to Differentiation

Slope of a line indicates the rate at which a line rises or falls

For a line, this rate (or slope) is the same at every point on the line

Tangent Line to a Graph

To determine the rate at which a graph rises or falls at a single point, you can find the slope of
the tangent line at the point.
In simple terms, the tangent line to the graph of a function at a point P is the line that best
approximates the graph at that point.

Slope of a Graph

Because a tangent line approximates the graph at a point, the problem of finding the slope of a
graph at a point becomes one of finding the slope of the tangent line at the point.
Example 1.1 Use the graph to approximate the slope of the graph of 𝑓 𝑥 = 𝑥2 at the point (1 ,1).
From the graph of 𝑓 𝑥 = 𝑥2, you can see that the tangent line at (1,1) rises approximately two units
for each unit change in x. Thus, the slope of the tangent line at (1,1) is given by