Slope and Derivative

 

Motivation:

 

                          We plot a function f(x) on the XY-plane as below.

                                 

                          When x=d,  the function is with the value f(d) and there is a corresponding point A(d,f(d)) on the XY-plane.  Similarly, when x=d+h , the function is with the value f(d+h) and there is also a corresponding point B(d+h, f(d+h)) .

 

                          Obviously, the two points A and B are on the curve of f(x) . So, the line passing A and B also crosses f(x) at least two points : A and B.   What is the slope of this line?  The slope m is

 

                                         

 

 

 

                         What if we gradually decrease h ?  The point B should gradually move along the curve toward A. When h is very very small,  point B is almost merging to point A. The line is almost the tangent line of the curve f(x) at x=d .  So, the slope of the tangent line of the curve f(x) at x=d  should be

 

                                       if this limit exists .

 

     

             If the limit exists, in general, we obtain a new function for the slope of the tangent line of f(x) at any point :

 

                               

 

                 And we denote this new function as   , known as derivative function of f(x). 

 

 

 

 

 

Definition ( Derivative Function )

            Let   f(x)  be a function of x such that  .  Then the derivative of  f(x) , denoted as   ,  is defined as

 

                                  

 

           And the domain of   is where the limit exists.  If y=f(x), we also denote this derivative function as

                                                            

Up to this moment, we just think it is a symbol to represent the derivative of f(x). Furthermore, if the limit exists when x=c,  then we say  f(x)  is differentiable at x=c .

 

           Please recall that the definition of the existence of  “limit” of a function at a point:  the limit from each direction should be the same.  So, to check if the derivative is well-defined at some points,  we should consider  h  to approach 0 from both directions in one-dimensional space.

 

 

 

Theorem :  Let  f(x)  be a function of x such that  .  If   f(x) is differentiable at x=c,  then  f(x)  is continuous at x=c .

 

 Proof:

                      The derivative of  f  exist at x=c .   Let’s say the limit is d at x=c. So,

 

                                          

 

                      In other words,  for any  >0 , there exists a corresponding  such that

 

                                 when  | h | <

 

                   d-  <   < d+       when  | h | <  

 

                Let   M = max { | d-  |, |d+  | } .   Then

                                 

                       | f(c+h) –f(c) | <  |h| M    <   M      when  | h | <

 

               That means   when  |h|  0,   f(c+h)  f(c) .  So, f(x) is continuous at x=c.

 

Note:   The converse of this theorem is not true.

           In other words,  if a function is continuous, it does not have to be differentiable.

 

Example:   Consider the following diagram:

                                 

                   Let’s try not to use any equation in this example.  With diagram like this, f(x) is continuous at O. But recall how we define the derivative function:  it is the slope of the line that you choose two points that are located very closely on the “curve” of f(x).  In this example, f(x) are composed of two “rays”.  So, if you follow the original definition of derivative to find the slope of such lines by pick two points near O, you will get many different answers.   Thus,  f(x) is not differentiable near O .

 

 

 

 

Derivative of  f(x)=xⁿ

 

                Given x, let’s find out the derivative of  f(x)=xⁿ :

 

                   f(x+h) – f(x) =  (x+h)ⁿ – xⁿ

                                        = ( xⁿ + n x^n-1 h +  x^n-2h² +  x^n-3h³ + … + hⁿ ) - xⁿ

                                        = nx^n-1h +  x^n-2h² +  x^n-3h³ + … + hⁿ

            So,

                      = nx^n-1 +  x^n-2h +  x^n-3h² + … + h^n-1

 

                       = nx^n-1

 

            Thus,      = nx^n-1  .

 

 

Example:  f(x)=3x²  .  Then  = 6x .

 

 

Ratio of Two Infinitesimally Small Quantities

 

                                          

 

 

                          At the beginning of this section,  the relationship between the slope of tangent line and derivative function was mentioned.  And the notation  for the derivative of y=f(x) was also introduced.  Up to this moment,   is just only a notation. But when we link it to the slope of the tangent line, it can be considered as the ratio of two infinitesimally small variables.  As shown in the diagram above,  the slope of the tangent line of f(x) at x=c  is

 

                                          

                    And remember it is a fixed value .

 

 

 

                  Comparing with the increased quantity  for f(x) moving from x=c to x=c+h, the quantity q increased along the tangent line is

 

                                 

                                   = f(c+h) – f(c)

 

 

 

 

                  And we know that the slope of the tangent line is obtained by  and that is  .  When we think  as the ratio of two infinitesimally small variables,  obviously   is changed as  varies;  and this ratio also changes when the location is changed  .   So,  when   =h   around x=c ,  we might as well put

 

                                               = 

 

                 While h is very small,

 

                                 =

 

                 And sometimes, we can approximate  f(c+h)  as

 

                                       f(c+h) = f(c) +