2.3
Differentiation Rules for Products, Quotients,
Secants, and Tangents
Product Rule
d
[f ( x ) ! g ( x ) ] = f ( x ) g ' ( x ) + g ( x ) f ' ( x )
dx
The 1st times the der. of the 2nd + the 2nd times
the der. of the 1st.
uv’+vu’
Ex.
f(x) = (3x – 2x2)(5 + 4x)
f’(x) = (3x – 2x2)(4) + (5 + 4x)(3 – 4x)
f’(x) = 12x – 8x2 + 15 – 20x + 12x – 16x2
f’(x) = -24x2 + 4x + 15
d
[ x sin x] = x cos x + sin x (1) = x cos x + sin x
dx
y = 2x cos x – 2 sin x
y’ = (2x)(-sin x) + cos x (2) - 2 cos x = -2x sin x
Quotient Rule
g(x) f '(x) - f (x)g'(x)
d È f ( x) ˘
=
2
Í
˙
g(x)
dx Î g ( x) ˚
[ ]
Differentiate
2x - 4x + 3
y=
2 - 3x
2
y’ =
(2 – 3x) (4x – 4) - (2x2 – 4x + 3) (-3)
(2 - 3x)
2
2 + 20x – 8) – (- 6x2 + 12x – 9)
(-12x
y’ =
(2 - 3x)
2 + 8x + 1
-6x
y’ =
(2 - 3x)
2
2
1- cos x
y=
sin x
(sin x)(sin x) - (1 - cos x)(cos x)
y' =
2
(sin x)
sin x - cos x + cos x
y' =
2
sin x
2
1 - cos x
y' =
2
sin x
2
Derivatives of Trigonometric Functions
d
[tan x]= sec 2 x
dx
d
[cot x]= - csc 2 x
dx
d
[sec x]= sec x tan x
dx
d
[csc x]= - csc x cot x
dx
Differentiate
y = x – tan x
y = x sec x
dy
2
= 1- sec x
dx
y’ = x(sec x tan x) + (sec x)(1)
y’ = sec x(x tan x + 1)