Benginning Calculus Lecture notes 4 - rules

Beginning Calculus
- Rules of Di¤erentiation -
Shahrizal Shamsuddin Norashiqin Mohd Idrus
Department of Mathematics,
FSMT - UPSI
(LECTURE SLIDES SERIES)
VillaRINO DoMath, FSMT-UPSI
(D3) Rules of Di¤erentiation 1 / 17

Rules of Di¤erentiation Derivatives of Trigonometric Functions
Learning Outcomes
State and apply the rules of di¤erentiation to evaluate derivatives.
State and apply the derivatives of trigonometric functions.

The Constant Rule
If f (x) = c for any constant c, then
f 0
(x) = 0 (1)
Proof:
f 0
(x) = lim
∆x!0
c c
∆x
= lim
∆x!0
0 = 0

The Power Rule
If f (x) = xn, with n 2 Z+, then
f 0
(x) = nxn 1
(2)

The Constant Multiple Rule
If c is a constant and f is a di¤erentiable function, then
(cf )0
(x) = cf 0
(x) (3)
Proof:
(cf )0
(x) = lim
∆x!0
(cf ) (x + ∆x) (cf ) (x)
∆x
= lim
∆x!0
c
f (x + ∆x) f (x)
∆x
= c lim
∆x!0
f (x + ∆x) f (x)
∆x
= c f 0
(x)

The Sum and Di¤erence Rules
If f and g are di¤erentiable functions, then
(f g)0
(x) = f 0
(x) g0
(x) (4)

Example
d
dx
5
p
x
10
x2
+
1
2
p
x
=
d
dx
5
p
x
d
dx
10
x2
+
d
dx
1
2
p
x
= 5
1
2
x 1/2
10 ( 2) x 3
+
1
2
1
2
x 3/2
=
5
2
p
x
+
20
x3
1
4x3/2

The Product Rule
(fg)0
(x) = f 0
(x) g (x) + g0
(x) f (x) (5)

Example
d
dx
h
2x3
+ 3 x4
2x
i
= x4
2x
d
dx
2x3
+ 3 + 2x3
+ 3
d
dx
x4
2x
= x4
2x (6x) + 2x3
+ 3 4x3
2

The Quotient Rule
f
g
0
(x) =
f 0 (x) g (x) g0 (x) f (x)
[g (x)]2
(6)

Example
d
dx
x2 + x 2
x3 + 6
=
x3 + 6
d
dx
x2 + x 2 x2 + x 2
d
dx
x3 + 6
(x3 + 6)
2
=
x3 + 6 (2x + 1) x2 + x 2 3x2
(x3 + 6)
2

Derivative of Sin x
d
dx
(sin x) = lim
∆x!0
sin (x + ∆x) sin x
∆x
= lim
∆x!0
sin x cos ∆x + cos x sin ∆x sin x
∆x
= lim
∆x!0
sin x
cos ∆x 1
∆x
+ cos x
sin ∆x
∆x
= lim
∆x!0
[sin x (0) + cos x (1)] = cos x
Note: lim
∆x!0
cos ∆x 1
∆x
= 0, and lim
∆x!0
sin ∆x
∆x
= 1. These were shown
geometrically in previous Limits and Continuity.

Derivative of Cos x
d
dx
(cos x) = lim
∆x!0
cos (x + ∆x) cos x
∆x
= lim
∆x!0
cos x cos ∆x sin x sin ∆x cos x
∆x
= lim
∆x!0
cos x (cos ∆x 1) sin x sin ∆x
∆x
= lim
∆x!0
cos x
cos ∆x 1
∆x
sin x
sin ∆x
∆x
= lim
∆x!0
cos x (0) sin x (1) = sin x

Remark
d
dx
(cos x)
x=0
= lim
∆x!0
cos ∆x 1
∆x
= 0
d
dx
(sin x)
x=0
= lim
∆x!0
sin ∆x
∆x
= 1
Derivatives of sine and cosine at x = 0 gives all the values of
d
dx
(sin x)
and
d
dx
(cos x) .

Derivative Formulas For Trigonometry
d
dx
(sin x) = cos x (7)
d
dx
(cos x) = sin x
d
dx
(tan x) = sec2
x
d
dx
(sec x) = sec x tan x
d
dx
(csc x) = csc x cot x
d
dx
(cot x) = csc2
x

Example
d
dx
(tan x) = sec2 x

Example
d
dx
1 + tan x
1 tan x
=
2 sec2 x
(1 tan x)2

Benginning Calculus Lecture notes 4 - rules

More Related Content

What's hot (20)

Viewers also liked (10)

Similar to Benginning Calculus Lecture notes 4 - rules (20)

Recently uploaded (20)

Benginning Calculus Lecture notes 4 - rules