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.