'- - File: IV_Der1.txt 

' We cover three examples.
' Ex. #1: Find the first and second derivative evaluated at x = 3.

a = -1.35
F{x} = x^5  + Sin[3*x + a]  + 5^x
Da{x} = Diff[F,x]                                ' 1st derivative.
Dn{x} = Diff[F,x | 3]  :  m2 = Da{3} 

 ' CANNOT set m2 directly to Diff[F,x |3] because a constant cannot be assigned 
'       to any expression containing a parameter.
'      Dn{x} results in a numerical only quantity and should equal m2, or Da{3}
'      Dn{x} is a constant and should equal m2 or Da{3} by setting <x = 3> assignment.

Db{x} = Diff[F,x,2]                             ' 2nd derivative.
Dm{x} = Diff[F,x,2 | 3]  : m3 = Db{3}

' - - - - - - - -  - - - - - - - - - - - - - - A trig identitiy check - - - - - - - - - - - - - - - - - - - - - - - - - - -
' Ex. #2: Verify that a trig identity is unity & its derivative = 0.

A{x} = Sin[F{x}]^2 + Cos[F{x}]^2
Ad{x} = Diff[A,x]                                 ' Should be zero.
' The complete expresson for Ad{x} can be printed out in landscape mode on your printer.
' However, since we are concerned more with graphs, numerical results become necessary
'     to verify.
' - - - - - -  - - - - - - - - - - - Sqrt[x] versus x ^ 0.5 power concerns - - - - - - - - - - - - - - - - - - - 
' Ex. #3: Caution using Sqrt[x] vs x^0.5 power.

R{x} = Sqrt[x] - x^0.5                          ' Should be zero
Rd{x} = Diff[R,x]                                 ' Should be zero.
' The resuts are as expected. Higher derivatives could result in output expressions when
'      using the Sqrt[x] rather than x^0.5 in your statements:
Re{x} = Diff[R,x,4]                              ' Upto 9th derivative. Need More? Set output to another
                                                             '     function and continue--NOT recommended due to
                                                             '     exceeding 60,000 symbols for the program to handle.
' If you use 9th derivative, the symbol table is exceeded, and the program may crash. Thus,
'     use x^0.5 if such a need exists beyond the 3rd or 4th derivative.

' In summary, please save your work if you plan on working with higher derivatives. The plots
'     covered in later examples rarely exceed the 2nd derivative in applications.

<x = 3>

