Definition.   Let h be not 0.  The difference f(x + h) - f(x) is called the increment of f from x to x+h, and is

denoted by f =  f(x + h) - f(x).  The product f '(x) h is called the differential of f at x with increment h, and

is denoted by  df = f'(x)h.  

Application #1:  Use a  differential  to estimate the change in f(x) = x^(2/5) if:

• (A) x is increased from 32 to 34  --  this is given by df = f ' (x) h, where h = 34 - 32.

• (B) x is increased from 1 to 9/10  --  this is given by df = f ' (x) h, where h = 1- 9/10.

(Step by Step Solution -- part A) :   Here x = 32 and h = 34 -x.
 

1.  F(x) := x^(2/5);     returns  RESULT
2.  define(G(x) , diff(F(x),x));     returns  RESULT
3.  define (df(x,h), G(x) * h)     returns  RESULT
4.  h  = 34 - 32;     returns  RESULT
5.  Sodf(32,34-32),numer=true;     returns RESULT,

(One Step Solution -- part A) :   Here x = 32 and h = 34 -x.
 
 

• block(F(x):= x^(2/5), define(G(x),diff(F(x),x)), define (df(x,h), G(x)*h), df(32,2))     returns  RESULT

1.  Redefine h by 9/10-1;     returns  RESULT
2.   Hence df(1,9/10-1);     returns  RESULT