The Derivative

Introduction:

Let F be a fuction, and P(X,F(X)) a point on its graph. What is the slope of the tangent line, if any, to the graph at that point?

Answer: We choose a small nonzero number H and on the graph mark the point Q(X+H,F(X+H)). Draw a secant line that passes through P and Q. As H approach zero from the right and/or from the left this line approach the limiting position of these secants. Since the secant lines have slopes:

(1) ,

then the tangent line (if it exists) have slope:

(2) .

The applet -- instructions:

In this applet you can visualize the question aforementioned. It is given a function F(X) = -X/2 * sin(X), a point P(X,F(X)), and Q(X+H,F(X+H)). You can move P or change the value of H by pressing the respective button and dragging the points with the mouse.

• Move P and see the slope of the secant (or tangent) line changing.
• Now change the value of h by dragging Q closer and closer to P till H equals 0. In this case, since F(X) is continuous, the tangent line exists for all (X,F(X)) in the graph and so does the lim defined in (2). Now with H = 0, move P and check the values of the lim defined in (2).

Definition:

A function F is said to be differentiable at X iff

(2)  exists.

If this limit exists, it is called the derivative of F at X denoted by F'(x). We say that F is differentiable function if it is differentiable at each X in its domain.

Examples:

1. Is f(x) = 1/x differentiable?
    
   Solution:
   

   Therefore f(x) = 1/x is differentiable at all real x except when x = 0.

    

2. Find f'(2) given that f(x) = 5 - x³.
    
   Solution: Let x = 2
    

   f'(2)	=
   	=
   	=
   	=