Page 251 - DMTH201_Basic Mathematics-1

Basic Mathematics – I

Notes 9.2 General Theorems on Derivatives : (Without Proof)

k
d ( )
Theorem 1: 0 where k is a constant.
dx
d d
x
x
Theorem 2: kf ( ) k f ( ) where k is a constant and ( )f x is a function of x.
dx dx
or
d du
ku k
dx dx
d d d
x
x
x
x
Theorem 3: f ( ) g ( ) f ( ) g ( )
dx dx dx
or
d du dv
(u v )
dx dx dx
d d d
x
x
x
x
Theorem 4: f ( ) g ( ) f ( ) g ( )
dx dx dx
or
d du dv
u v
dx dx dx
d d d
x
x
x
x
x
x
Theorem 5: f ( ) g ( ) f ( ) g ( ) g ( ) f ( )
dx dx dx
This is called the product rule.
d dv du
or (uv ) u v
dx dx dx
This can be remembered as
d
I function II function
dx
d d
I function II function II function I function
dx dx
d dw dv du
(uvw ) uv uw vw
Notes dx dx dx dx
d d
g ( ) x f ( ) x f ( ) x g ( ) x
d f ( ) x dx dx
Theorem 6: 2
dx g ( ) g ( )
x
x
This is called the Quotient Rule.
du dv
v u
d u
or dx dx
dx v v 2

244 LOVELY PROFESSIONAL UNIVERSITY