Calculus and Differential Equation

CALCULUS AND DIFFERENTIAL EQUATIONS (22MATS11)

Module 1 - Calculus

Prerequisites:
Trigonometry

Pythagorean identities Reciprocal ratios
sin2 𝜃 + cos 2 𝜃 = 1 1
cosec 𝜃 =
sin 𝜃
sec 2 𝜃 − tan2 𝜃 = 1 1
sec 𝜃 =
cos 𝜃
𝑐𝑜sec 2 𝜃 − cot 2 𝜃 = 1 1
cot 𝜃 =
tan 𝜃

Sum formulas Difference formulas
sin(𝑥 + 𝑦) = sin 𝑥 cos 𝑦 + cos 𝑥 sin 𝑦 sin(𝑥 − 𝑦) = sin 𝑥 cos 𝑦 − cos 𝑥 sin 𝑦
cos(𝑥 + 𝑦) = cos 𝑥 cos 𝑦 − cos 𝑥 cos 𝑦 cos(𝑥 − 𝑦) = cos 𝑥 cos 𝑦 + cos 𝑥 cos 𝑦
tan 𝑥 + tan 𝑦 tan 𝑥 − tan 𝑦
tan(𝑥 + 𝑦) = tan(𝑥 − 𝑦) =
1 − tan 𝑥 tan 𝑦 1 + tan 𝑥 tan 𝑦
Double angle formulas Triple angle formulas
sin 2𝑥 = 2 sin 𝑥 cos 𝑥 sin 3𝑥 = 3 sin 𝑥 − 4 sin3 𝑥
cos 2𝑥 = cos 2 𝑥 − sin2 𝑥 cos 3𝑥 = 4 cos 3 𝑥 − 3 cos 𝑥
2 tan 𝑥 3 tan 𝑥 − tan3 𝑥
tan 2𝑥 = tan 3𝑥 =
1 − tan2 𝑥 1 − 3 tan2 𝑥
Half angle formulas Tangent formulas
1 2 tan 𝑥
sin2 𝑥 = (1 − cos 2𝑥) sin 2𝑥 =
2 1 + tan2 𝑥
1 1 − tan2 𝑥
cos2 𝑥 = (1 + cos 2𝑥) cos 2𝑥 =
2 1 + tan2 𝑥
1 − cos 2𝑥 2 tan 𝑥
tan2 𝑥 = tan 2𝑥 =
1 + cos 2𝑥 1 − tan2 𝑥

Standard angle formulas ASTC Rule
𝜃 𝑜° 30° 45° 60° 90°
sin 𝜃 0 1 1 √3 90°
1
2 √2 2 (−, +) (+, +)
√3 1 1
cos 𝜃 1 0 S A
2 √2 2
180° 0°
1
tan 𝜃 0 1 √3 ∞ T C
√3
(−, −) (+, −)
270°

Dr. Narasimhan G. RNSIT 1

,𝐍𝐨𝐭𝐞:
𝑥 𝑥 𝑥 2 𝜋 1 + tan 𝑥
2 sin2 = 1 − cos 𝑥 (cos + sin ) = 1 + sin 𝑥 tan ( + 𝑥) =
2 2 2 4 1 − tan 𝑥
𝑥 𝑥 𝑥 2 𝜋 1 − tan 𝑥
2 cos2 = 1 + cos 𝑥 (cos − sin ) = 1 − sin 𝑥 tan ( − 𝑥) =
2 2 2 4 1 + tan 𝑥


Same ratio formulas:
sin(−𝜃) = − sin 𝜃 sin(2𝜋 − 𝜃) = −sin 𝜃 sin(𝜋 − 𝜃) = sin 𝜃 sin(𝜋 + 𝜃) = − sin 𝜃

cos(−𝜃) = cos 𝜃 cos(2𝜋 − 𝜃) = cos 𝜃 cos(𝜋 − 𝜃) = − cos 𝜃 cos(𝜋 + 𝜃) = − cos 𝜃

tan(−𝜃) = − tan 𝜃 tan(2𝜋 − 𝜃) = − tan 𝜃 tan(𝜋 − 𝜃) = − tan 𝜃 tan(𝜋 + 𝜃) = tan 𝜃

cot(−𝜃) = − cot 𝜃 cot(2𝜋 − 𝜃) = − cot 𝜃 cot(𝜋 − 𝜃) = − cot 𝜃 cot(𝜋 + 𝜃) = cot 𝜃

sec(−𝜃) = sec 𝜃 sec(2𝜋 − 𝜃) = sec 𝜃 sec(𝜋 − 𝜃) = − sec 𝜃 sec(𝜋 + 𝜃) = − sec 𝜃

cosec(−𝜃) = − cosec 𝜃 cosec(2𝜋 − 𝜃) = − cosec 𝜃 cosec(𝜋 − 𝜃) = cosec 𝜃 cosec(𝜋 + 𝜃) = − cosec 𝜃

(IV quadrant) Cos +ve (IV quadrant) Cos +ve (II quadrant) Sin +ve (III quadrant) Tan +ve




Co ratio formulas:
𝜋 𝜋 3𝜋 3𝜋
sin ( − 𝜃) = cos 𝜃 sin ( + 𝜃) = cos 𝜃 sin ( − 𝜃) = − cos 𝜃 sin ( + 𝜃) = −cos 𝜃
2 2 2 2
𝜋 𝜋 3𝜋 3𝜋
cos ( − 𝜃) = sin 𝜃 cos ( + 𝜃) = − sin 𝜃 cos ( − 𝜃) = − sin 𝜃 cos ( + 𝜃) = sin 𝜃
2 2 2 2
𝜋 𝜋 3𝜋 3𝜋
tan ( − 𝜃) = cot 𝜃 tan ( + 𝜃) = − cot 𝜃 tan ( − 𝜃) = cot 𝜃 tan ( + 𝜃) = − cot 𝜃
2 2 2 2
3𝜋 3𝜋
𝜋 𝜋 cot ( − 𝜃) = tan 𝜃 cot ( + 𝜃) = −tan 𝜃
cot ( − 𝜃) = tan 𝜃 cot ( + 𝜃) = − tan 𝜃 2 2
2 2
3𝜋 3𝜋
𝜋 𝜋 sec ( − 𝜃) = − cosec 𝜃 sec ( + 𝜃) = 𝑐𝑜 sec 𝜃
sec ( − 𝜃) = cosec 𝜃 sec ( + 𝜃) = −𝑐𝑜 sec 𝜃 2 2
2 2
𝜋 𝜋 3𝜋 3𝜋
cosec ( − 𝜃) = sec 𝜃 cosec ( + 𝜃) = sec 𝜃 cosec ( − 𝜃) = − sec 𝜃 cosec ( + 𝜃) = − sec 𝜃
2 2 2 2
(I quadrant) All +ve (II quadrant) Sin +ve (III quadrant) Tan +ve (IV quadrant) Cos +ve




Dr. Narasimhan G. RNSIT 2

, Differentiation of some standard functions

Non Trigonometric Trigonometric Hyperbolic functions Inverse functions
functions functions

(𝒌)′ = 𝟎 (𝒔𝒊𝒏 𝒙)′ = 𝒄𝒐𝒔 𝒙 (𝒔𝒊𝒏𝒉 𝒙)′ = 𝒄𝒐𝒔𝒉 𝒙 𝟏
(𝒔𝒊𝒏−𝟏 𝒙)′ =
√𝟏 − 𝒙𝟐

(𝒙𝒏 )′ = 𝒏 𝒙𝒏−𝟏 (𝒄𝒐𝒔 𝒙)′ = −𝒔𝒊𝒏 𝒙 (𝒄𝒐𝒔𝒉 𝒙)′ = 𝒔𝒊𝒏𝒉 𝒙 𝟏
(𝒄𝒐𝒔−𝟏 𝒙)′ = −
√𝟏 − 𝒙𝟐

𝟏 (𝒕𝒂𝒏 𝒙)′ = 𝒔𝒆𝒄𝟐 𝒙 (𝒕𝒂𝒏𝒉 𝒙)′ = 𝒔𝒆𝒄𝒉𝟐 𝒙 𝟏
(√𝒙)′ = (𝒕𝒂𝒏−𝟏 𝒙)′ =
𝟐√𝒙 𝟏 + 𝒙𝟐

𝟏 (𝒄𝒐𝒕 𝒙)′ = − 𝒄𝒐𝒔𝒆𝒄𝟐 𝒙 (𝒄𝒐𝒕𝒉 𝒙)′ = − 𝒄𝒐𝒔𝒆𝒄𝒉𝟐 𝒙 𝟏
(𝒍𝒐𝒈 𝒙)′ = (𝒄𝒐𝒕−𝟏 𝒙)′ = −
𝒙 𝟏 + 𝒙𝟐

(𝒆𝒙 )′ = 𝒆𝒙 (𝒔𝒆𝒄 𝒙)′ = 𝒔𝒆𝒄 𝒙. 𝒕𝒂𝒏 𝒙 (𝒔𝒆𝒄𝒉 𝒙)′ = −𝒔𝒆𝒄𝒉 𝒙. 𝒕𝒂𝒏𝒉 𝒙 𝟏
(𝒔𝒆𝒄−𝟏 𝒙)′ =
𝒙√𝒙𝟐 − 𝟏

(𝒂𝒙 )′ = 𝒂𝒙 𝒍𝒐𝒈 𝒂 (𝒄𝒐𝒔𝒆𝒄 𝒙)′ (𝒄𝒐𝒔𝒆𝒄𝒉 𝒙)′ 𝟏
(𝒄𝒐𝒔𝒆𝒄−𝟏 𝒙)′ = −
= −𝒄𝒐𝒔𝒆𝒄 𝒙. 𝒄𝒐𝒕 𝒙 = −𝒄𝒐𝒔𝒆𝒄𝒉 𝒙. 𝒄𝒐𝒕𝒉 𝒙 𝒙√𝒙𝟐 − 𝟏


Rules of differentiation

1. (𝒌𝒖)′ = 𝒌𝒖′ 3. (𝒖𝒗)′ = 𝒖𝒗′ + 𝒗𝒖′
𝒖 ′ 𝒗𝒖′ −𝒖𝒗′
2. (𝒖 ± 𝒗)′ = 𝒖′ ± 𝒗′ 4. (𝒗) = 𝒗𝟐




Dr. Narasimhan G. RNSIT 3