4-PrincipCalculus Page 110 Friday, March 12, 2004 12:39 PM

110

EXHIBIT 4.6

Commonly Used Rules of Derivation

Function

Standard Notation

Infinitesimal Notation

Termwise dif-

hx

()

+

bg x

x

x

x

()

=

af x

()

h

′()

=

af

′()

+

bg

′()

or

dh

df

b

dg

-------

=

-------a

+

------

• 
  

ferentiation

dx

dx

dx

Product rule

hx

()

gx

x

x

()

+

fx

x

-------

=

------

+

------

• 
  

()

=

fx

()

h

′()

=

f′()

gx

()

g′()

or

dh

df

• gf

dg

dx

dx

dx

Quotient rule

1

g′()

or

dh

1

dg

x

hx

x

()

=

-----------

h

′()

= –

------------------

• 
  

-------

= –

------------------

---

• 
  

()

(gx

()

(^2) )
dx
gx
()
(^2) )
dx
(gx
Chain rule
hx
( ()
)
h′()

f ′(
gx
x

=

---

• 
  

()

=

fgx

x

()

)g

′()

dh

df

dg

dx

dg

dx