Geometric Interpretation of the Derivative

Graph of the Derivative

When h approaches zero, the point Q approaches point P. At this time, the secant line begins to resemble the tangent to the function f(x) at Point P, and thus the angle α tends to be β.

Derivative

Derivative Graph

The slope of the tangent to the curve at a point is equal to the derivative of the function at that point.

mt = f'(a)

Examples

Given the parabola f(x) = x2, find the points where the tangent line is parallel to the bisector of the first quadrant.

The bisector of the first quadrant has the equation y = x, so its slope is m = 1.

Since the two lines are parallel they have the same slope, so:

f'(a) = 1.

Since the slope of the tangent to the curve equals the derivative at x = a.

Derivative

Derivative

Derivative

Derivative Graph