The Definition of the Derivative LESSON 3 OF 20. Deriving the Formula You can use the coordinates in reverse order and still get the same result. It doesn’t.

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Presentation transcript:

The Definition of the Derivative LESSON 3 OF 20

Deriving the Formula You can use the coordinates in reverse order and still get the same result. It doesn’t matter in which order you do the subtraction as long as you’re consistent.

Calculating Slope Given Two Points on a Line

Looking at Slope Graphically

Looking at Slope Graphically (cont.) Don’t panic – only the notation has changed.

The Slope of a Curve  Suppose that we wanted to find the slope of a curve instead. Here the slope formula no longer works because the distance from one point to the other is along a curve, not a straight line.  But we could find the approximate slope if we took the slope of the line between the two points. This is called the secant line.

The Secant and the Tangent  As you can see, the farther apart the two points are, the less the slope of the line corresponds to the slope of the curve.  Conversely, the closer the two points are, the more accurate the approximation is.

The Secant and the Tangent (cont.)

The Definition of the Derivative

Example 1

Example 2

Example 3

Differentiability  One of the important requirements for the differentiability of a function is that the function be continuous.  But, even if a function is continuous at a point, the function is not necessarily differentiable there.