FORMULA SHEET FOR DBM20023 EXPONENTS AND LOGARITHMS LAW OF EXPONENTS

LAW OF LOGARITHMS

1.

𝑎𝑚 × 𝑎𝑛 = 𝑎𝑚+𝑛

8.

log𝑎 𝑎 = 1

2.

𝑎𝑚 = 𝑎𝑚−𝑛 𝑎𝑛

9.

log𝑎 1 = 0

3.

(𝑎𝑚)𝑛 = 𝑎𝑚×𝑛

10.

log𝑎 𝑏 =

4.

𝑎0 = 1

11.

log𝑎 𝑀𝑁 = log𝑎 𝑀 + log𝑎 𝑁

5.

𝑎−𝑛 =

12.

log𝑎

6.

𝑎 𝑛 = ( √𝑎)𝑚

13.

log𝑎 𝑁𝑃 = 𝑃 log𝑎 𝑁

7.

(𝑎𝑏)𝑛 = 𝑎𝑛𝑏𝑛

14.

𝑁 = 𝑎𝑥 ⇔ log𝑎 𝑁 = 𝑥

𝑚

1 𝑎𝑛

,𝑎≠ 0

𝑛

log𝑐 𝑏 log 𝑎𝑐

𝑀 = log𝑎 𝑀 − log𝑎 𝑁 𝑁

DIFFERENTIATION 1.

3.

5.

7.

9.

11.

13.

15.

𝑑

𝑑𝑥

[𝑘] = 0

𝑑 𝑑𝑥 𝑑 𝑑𝑥 𝑑 𝑑𝑥 𝑑 𝑑𝑥 𝑑 𝑑𝑥 𝑑 𝑑𝑥

, k is constant

2.

[𝑓(𝑥) ± 𝑔(𝑥)] = 𝑓′(𝑥) ± 𝑔′(𝑥)

4.

[𝑎𝑒𝑢(𝑥)] = 𝑎𝑒𝑢(𝑥) · 𝑢′(𝑥)

6.

[𝑎 sin 𝑢(𝑥)] = 𝑎 cos 𝑢(𝑥) · 𝑢′(𝑥)

8.

[𝑎 cos 𝑢(𝑥)] = −𝑎 sin 𝑢(𝑥) · 𝑢′(𝑥)

10.

[𝑎 tan 𝑢(𝑥)] = 𝑎 𝑠𝑒𝑐2 𝑢(𝑥) · 𝑢′(𝑥)

12.

[𝑦{𝑢(𝑥)}] =

𝑑𝑦

×

𝑑𝑢

𝑑𝑢 𝑑𝑥 𝑑𝑢 − 𝑢 𝑑𝑣) (𝑣 𝑑𝑥 𝑑 𝑢(𝑥) 𝑑𝑥 [ ]= 2 (𝑣) 𝑑𝑥 𝑣(𝑥)

[Chain Rule]

14.

[Quotient Rule]

16.

𝑑 𝑑𝑥 𝑑

[ 𝑎𝑥𝑛] = 𝑛𝑎𝑥𝑛−1

[Power Rule]

[𝑎{𝑢(𝑥)}𝑛] = 𝑛𝑎{𝑢(𝑥)}𝑛−1 · 𝑢′(𝑥) [Composite]

𝑑𝑥

𝑑 𝑑𝑥 𝑑 𝑑𝑥 𝑑

𝑑 𝑑𝑥 𝑑

𝑎 𝑢( 𝑥 )

· 𝑢′(𝑥)

[𝑎 sin𝑛 𝑢(𝑥)] = 𝑎 cos 𝑢(𝑥) · sin𝑛−1 𝑢(𝑥) · 𝑢′(𝑥)

[𝑎 cos𝑛 𝑢(𝑥)] = −𝑎 sin 𝑢(𝑥) · cos𝑛−1 𝑢(𝑥) · 𝑢′(𝑥)

𝑑𝑥

𝑑𝑥

[𝑎ln 𝑢(𝑥)] =

[𝑎 tan𝑛 𝑢(𝑥)] = 𝑎 tan𝑛−1 𝑢(𝑥) · 𝑠𝑒𝑐2 𝑢(𝑥) · 𝑢′(𝑥)

[𝑢(𝑥) · 𝑣(𝑥)] = 𝑢

𝑑𝑦(𝑡) 𝑑𝑥(𝑡)

=

𝑑𝑦(𝑡) 𝑑𝑡

×

𝑑𝑡 𝑑𝑥(𝑡)

𝑑𝑣 𝑑𝑥

+𝑣

𝑑𝑢 𝑑𝑥

[Product Rule]

[Parametric Equation]

INTEGRATION 1.

∫ 𝑘 𝑑𝑥 = 𝑘𝑥 + 𝑐

, k is constant

3.

∫[𝑓(𝑥) ± 𝑔(𝑥)] 𝑑𝑥 = ∫ 𝑓(𝑥) 𝑑𝑥 ± ∫ 𝑔(𝑥) 𝑑𝑥

∫ 𝑎𝑥 𝑛𝑑𝑥 =

4.

∫ 𝑎[𝑢(𝑥)]𝑛𝑑𝑥 =

6.

∫

8.

∫ 𝑎 𝑠𝑖𝑛 𝑢( 𝑥) 𝑑𝑥 = −

10.

∫ 𝑎 𝑠𝑒𝑐2 𝑢( 𝑥) 𝑑𝑥 =

𝑏

5.

∫ 𝑓(𝑥)𝑑𝑥 = 𝐹(𝑏) − 𝐹(𝑎) 𝑎

𝑒𝑢(𝑥)

7.

∫ 𝑒𝑢(𝑥) 𝑑𝑥 =

9.

∫ 𝑎 𝑐𝑜𝑠 𝑢(𝑥)𝑑𝑥 =

𝑢′(𝑥)

+𝑐 𝑎 𝑠𝑖𝑛 𝑢(𝑥) 𝑢 ′( 𝑥 )

+𝑐

𝑎𝑥𝑛+1 + 𝑐 ; {𝑛 ≠ −1} ( 𝑛 + 1)

2.

𝑎[𝑢(𝑥)]𝑛+1 (𝑛 + 1) · 𝑢′(𝑥)

𝑎 𝑎 ln[𝑢(𝑥)] 𝑑𝑥 = [𝑢(𝑥)]𝑛 𝑢′(𝑥)

1.

𝑐𝑜𝑠2 𝜃 + 𝑠𝑖𝑛2 𝜃 = 1

2.

1 + 𝑡𝑎𝑛2 𝜃 = 𝑠𝑒𝑐2 𝜃

3.

1 + 𝑐𝑜𝑡2𝜃 = 𝑐𝑜𝑠𝑒𝑐2 𝜃

4.

sin 2𝜃 = 2 sin 𝜃 cos 𝜃

6.

tan 2𝜃 =

8.

cot 𝜃 =

10.

𝑐𝑜𝑠𝑒𝑐 𝜃 =

5. 7. 9.

cos 2𝜃 = 2 𝑐𝑜𝑠2𝜃 − 1 = 1 − 2 𝑠𝑖𝑛2𝜃 = 𝑐𝑜𝑠2𝜃 − 𝑠𝑖𝑛2𝜃 sin 𝜃 tan 𝜃 = cos 𝜃 sec 𝜃 =

1 cos 𝜃

2 tan 𝜃 1 − 𝑡𝑎𝑛2 𝜃

cos 𝜃 sin 𝜃

=

1 tan 𝜃

1 sin 𝜃

AREA UNDER CURVE 𝑏

1.

𝑏

𝐴𝑥 = ∫ 𝑦 𝑑𝑥

2.

𝑎

𝐴𝑦 = ∫ 𝑥 𝑑𝑦 𝑎

VOLUME UNDER CURVE 𝑏

1.

𝑉𝑥 = 𝜋 ∫ 𝑦2 𝑑𝑥 𝑎

𝑏

2.

𝑉𝑦 = 𝜋 ∫ 𝑥2 𝑑𝑦 𝑎

INTEGRATION BY PARTS ∫ 𝑢 𝑑𝑣 = 𝑢𝑣 − ∫ 𝑣 𝑑𝑢

𝑢′(𝑥)

𝑎 𝑡𝑎𝑛 𝑢(𝑥)

IDENTITY TRIGONOMETRY

; {𝑛 ≠ −1}

; {𝑛 = 1}

𝑎 𝑐𝑜𝑠 𝑢(𝑥)

𝑢′(𝑥)

+𝑐

+𝑐

+𝑐