

                       DIFFERENTIAL CALCULUS


     A function f(x) is continuous at x = a, if and only if

                       lim f(x) = f(a).
                       x─>a  

Limits of continuous functions can be evaluated by substitution.  If a  O/O 
or  ∞/∞ form results then algebraic simplification must precede
substitution.

Given that  lim f(x)  and  lim g(x)  exist, the following are true: 
            x->a           x->a

     1.  lim (f(x) ± g(x))  =  lim f(x) ± lim g(x)
         x->a                  x->a       x->a

     2.  lim (f(x) ∙ g(x))  =  lim f(x) ∙ lim g(x)
         x->a                  x->a       x->a
 
     3.  lim (f(x)/g(x))  =  lim f(x)/lim g(x)  provided  lim g(x) ╪ O
         x->a                x->a     x->a                x->a
 
     4.  lim f(x)  =  L <=> lim  f(x) = L  and  lim  f(x) = L .
         x->a               x->a+               x->a-

 

     The derivative of f(x) with respect to x evaluated at x  is
                                                            °

           df(x )         f(x + h) - f(x )          f(x) - f(x )
               °             °          °                     °
          ───────  =  lim ────────────────  =  lim  ────────────  
            dx        h->O        h            x->x    x - x
                                                   °        °

if the limit exists.  

Geometrically, the derivative gives the slope of the curve at the point
where it is evaluated.  


Rules for differentiation you will need are:


       d  ┌           ┐     df(x)     dg(x)
   1.  ── │f(x) ± g(x)│  =  ─────  ±  ─────
       dx └           ┘       dx        dx
 
       d  ┌           ┐          dg(x)          df(x)
   2.  ── │f(x) ∙ g(x)│  =  f(x) ─────  +  g(x) ─────
       dx └           ┘            dx             dx


                          df(x)          dg(x)
          ┌    ┐     g(x) ─────  -  f(x) ─────
       d  │f(x)│           dx             dx
   3.  ── │────│  =  ─────────────────────────
       dx │g(x)│           ┌    ┐2
          └    ┘           │g(x)│
                           └    ┘

         n
       dx        n-1                        d sin x
   4.  ───  =  nx                       5.  ───────  =  cos x
       dx                                     dx
 
       d cos x                              d tan x        2
   6.  ───────  =  -sin x               7.  ───────  =  sec x
         dx                                   dx
 
       d cot x         2                    d sec x
   8.  ───────  =  -csc x               9.  ───────  =  (sec x)(tan x)
         dx                                   dx
 
       d csc x
  1O.  ───────  =  -(csc x)(cot x)
         dx



The chain rule gives the derivative of f(u(x)) with respect to x as:    

        d f(u(x))     d f(u)    du
        ───────    =  ─────── ∙ ── .
           dx            du     dx 


This procedure can be used on formula 4-1O above to produce additional
formula.
   
The expression  y = f(x). is called explicit. Expressions which are
not explicit are called implicit, e.g. xy = 3.  To differentiate implicit
expressions, differentiate both sides of the expression with respect to the
independent variable.  Use the chain rule where needed.
