benginning calculus lecture notes 4 - rules
DESCRIPTION
Universiti Pendidikan Sultan IdrisTRANSCRIPT
Beginning Calculus- Rules of Differentiation -
Shahrizal Shamsuddin Norashiqin Mohd Idrus
Department of Mathematics,FSMT - UPSI
(LECTURE SLIDES SERIES)
VillaRINO DoMath, FSMT-UPSI
(D3) Rules of Differentiation 1 / 17
Rules of Differentiation Derivatives of Trigonometric Functions
Learning Outcomes
State and apply the rules of differentiation to evaluate derivatives.
State and apply the derivatives of trigonometric functions.
VillaRINO DoMath, FSMT-UPSI
(D3) Rules of Differentiation 2 / 17
Rules of Differentiation Derivatives of Trigonometric Functions
The Constant Rule
If f (x) = c for any constant c , then
f ′ (x) = 0 (1)
Proof:
f ′ (x) = lim∆x→0
c − c∆x
= lim∆x→0
0 = 0
VillaRINO DoMath, FSMT-UPSI
(D3) Rules of Differentiation 3 / 17
Rules of Differentiation Derivatives of Trigonometric Functions
The Power Rule
If f (x) = xn , with n ∈ Z+, then
f ′ (x) = nxn−1 (2)
VillaRINO DoMath, FSMT-UPSI
(D3) Rules of Differentiation 4 / 17
Rules of Differentiation Derivatives of Trigonometric Functions
The Constant Multiple Rule
If c is a constant and f is a differentiable function, then
(cf )′ (x) = cf ′ (x) (3)
Proof:
(cf )′ (x) = lim∆x→0
(cf ) (x + ∆x)− (cf ) (x)∆x
= lim∆x→0
cf (x + ∆x)− f (x)
∆x
= c lim∆x→0
f (x + ∆x)− f (x)∆x
= c · f ′ (x)
VillaRINO DoMath, FSMT-UPSI
(D3) Rules of Differentiation 5 / 17
Rules of Differentiation Derivatives of Trigonometric Functions
The Sum and Difference Rules
If f and g are differentiable functions, then
(f ± g)′ (x) = f ′ (x)± g ′ (x) (4)
VillaRINO DoMath, FSMT-UPSI
(D3) Rules of Differentiation 6 / 17
Rules of Differentiation Derivatives of Trigonometric Functions
Example
ddx
(5√x − 10
x2+
12√x
)=
ddx
(5√x)− ddx
(10x2
)+ddx
(12√x
)= 5
(12
)x−1/2 − 10 (−2) x−3 + 1
2
(−12
)x−3/2
=52√x+20x3− 14x3/2
VillaRINO DoMath, FSMT-UPSI
(D3) Rules of Differentiation 7 / 17
Rules of Differentiation Derivatives of Trigonometric Functions
The Product Rule
If f and g are differentiable functions, then
(fg)′ (x) = f ′ (x) g (x) + g ′ (x) f (x) (5)
VillaRINO DoMath, FSMT-UPSI
(D3) Rules of Differentiation 8 / 17
Rules of Differentiation Derivatives of Trigonometric Functions
Example
ddx
[(2x3 + 3
) (x4 − 2x
)]=
(x4 − 2x
) ddx
(2x3 + 3
)+(2x3 + 3
) ddx
(x4 − 2x
)=
(x4 − 2x
)(6x) +
(2x3 + 3
) (4x3 − 2
)
VillaRINO DoMath, FSMT-UPSI
(D3) Rules of Differentiation 9 / 17
Rules of Differentiation Derivatives of Trigonometric Functions
The Quotient Rule
If f and g are differentiable functions, then(fg
)′(x) =
f ′ (x) g (x)− g ′ (x) f (x)[g (x)]2
(6)
VillaRINO DoMath, FSMT-UPSI
(D3) Rules of Differentiation 10 / 17
Rules of Differentiation Derivatives of Trigonometric Functions
Example
ddx
(x2 + x − 2x3 + 6
)
=
(x3 + 6
) ddx
(x2 + x − 2
)−(x2 + x − 2
) ddx
(x3 + 6
)(x3 + 6)2
=
(x3 + 6
)(2x + 1)−
(x2 + x − 2
) (3x2)
(x3 + 6)2
VillaRINO DoMath, FSMT-UPSI
(D3) Rules of Differentiation 11 / 17
Rules of Differentiation Derivatives of Trigonometric Functions
Derivative of Sin x
ddx(sin x) = lim
∆x→0sin (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.
VillaRINO DoMath, FSMT-UPSI
(D3) Rules of Differentiation 12 / 17
Rules of Differentiation Derivatives of Trigonometric Functions
Derivative of Cos x
ddx(cos x) = lim
∆x→0cos (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→0cos x (0)− sin x (1) = − sin x
VillaRINO DoMath, FSMT-UPSI
(D3) Rules of Differentiation 13 / 17
Rules of Differentiation Derivatives of Trigonometric Functions
Remark
ddx(cos x)
∣∣∣∣x=0
= lim∆x→0
cos∆x − 1∆x
= 0
ddx(sin x)
∣∣∣∣x=0
= lim∆x→0
sin∆x∆x
= 1
Derivatives of sine and cosine at x = 0 gives all the values ofddx(sin x)
andddx(cos x) .
VillaRINO DoMath, FSMT-UPSI
(D3) Rules of Differentiation 14 / 17
Rules of Differentiation Derivatives of Trigonometric Functions
Derivative Formulas For Trigonometry
ddx(sin x) = cos x (7)
ddx(cos x) = − sin x
ddx(tan x) = sec2 x
ddx(sec x) = sec x tan x
ddx(csc x) = − csc x cot x
ddx(cot x) = − csc2 x
VillaRINO DoMath, FSMT-UPSI
(D3) Rules of Differentiation 15 / 17
Rules of Differentiation Derivatives of Trigonometric Functions
Example
ddx(tan x) = sec2 x
VillaRINO DoMath, FSMT-UPSI
(D3) Rules of Differentiation 16 / 17
Rules of Differentiation Derivatives of Trigonometric Functions
Example
ddx
(1+ tan x1− tan x
)=
2 sec2 x
(1− tan x)2
VillaRINO DoMath, FSMT-UPSI
(D3) Rules of Differentiation 17 / 17