Average Velocity A ball is thrown in the air.

The Derivative as a Function.

0:00 The Derivative as a Function.

2:27 Example, Sketch the graph of the derivative function using the graph of f(x), with six different graphs.

12:36 Theorem: Differentiation Implies Continuous and its Proof.

16:47 Theorem: Alternative Version. f(x) is not continuous implying that f(x) is not differentiable. With four examples showing when a function f(x) is not differentiable. Discontinuity, Corner points, Cusp " Sharp Points", Vertical Tangents.

23:11 Example Find the derivative of f(x) for the given function f(x)= 1/sqrt(x) and find the derivative at x= 25 in detail.

Introduction to Derivative.

0:00 Introduction.

This video covers the main two limit formulas applying the limit as x approaches a to the " slope formula" between two distinct points. Taking everything from the slope of the Secant to the slope of the Tangent at x = a. And that is the derivative of the function "f" at x = a. By applying the limit, we move from the Average velocity between two distinct points to the Instantaneous Velocity at a single point.

12:12 Example 1.

19:08 Example 2.

23:26 Example 3.

28:32 Example 4.