Lecture Note Differentiation
2.2 The Derivative of a constant
For any constant C,
() 0d
C
dxThat is, the derivative of a constant is zero.
2.3 The Constant Multiple Rule ....................................................................
For any constant C,
() ()ddf
Cf C Cf x
dx dx==′^
That is, the derivative of a constant time a function is equal to the constant times the
derivative of the function.
Example 2
Differentiate the function yx= 3 5
Solution:
You already know that ()^55
d 4
x x
dx=. Combining this with the constant multiple rule,you get () (^33554) () ( ) 3515
dd 4
x xx
dx dx
===x.
2.4 The Sum Rule .........................................................................................
() () ()
ddfdg
f gfx
dx dx dx±=± = ±′ g′ xThat is, the derivative of a sum is the sum of the individual derivatives.
Example 3
Differentiate the function yx x=+^253
Solution
(^) ()^2533 ()^2 ()^52
ddd 4
x xx xx 15
dx dx dx
+= + =+x
2.5 The Product Rule ....................................................................................
() () () () ()
ddfdg
fgg f fxgxgxfx
dx dx dx=+=′′+
That is, the derivative of a product is the first factor times the derivative of the second
plus the second factor times the derivative of the first.
Example 4
Differentiate the functionyx x=+^2 ( (^31) )
Solution
According to the product rule
()()() ()
()
222
2
2
31 31 31
312 3
92
dd
xx x x x x
dx dx dx
xxxdx x