FORMULA SHEET FOR DBM20023
EXPONENTS AND LOGARITHMS
LAW OF EXPONENTS
LAW OF LOGARITHMS
1.
𝑎𝑚×𝑎𝑛 = 𝑎𝑚+𝑛
8.
log𝑎𝑎 = 1
2.
𝑎𝑚
𝑎𝑛 = 𝑎𝑚−𝑛
9.
log𝑎1 = 0
3.
(𝑎𝑚)𝑛 = 𝑎𝑚×𝑛
10.
log𝑎𝑏 = log𝑐𝑏
log𝑐𝑎
4.
𝑎0= 1
11.
log𝑎𝑀𝑁 = log𝑎𝑀 + log𝑎𝑁
5.
𝑎−𝑛 = 1
𝑎𝑛 , 𝑎 ≠ 0
12.
log𝑎𝑀
𝑁 = log𝑎𝑀 − log𝑎𝑁
6.
𝑎𝑚
𝑛= (√𝑎
𝑛)𝑚
13.
log𝑎𝑁𝑃 = 𝑃 log𝑎𝑁
7.
(𝑎𝑏)𝑛 = 𝑎𝑛𝑏𝑛
14.
𝑁 = 𝑎𝑥⇔log𝑎𝑁 = 𝑥
DIFFERENTIATION
1.
𝑑
𝑑𝑥(𝑘)= 0, 𝑘 𝑖𝑠 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡
2.
𝑑
𝑑𝑥(𝑎𝑥𝑛)= 𝑎𝑛𝑥𝑛−1 [𝑃𝑜𝑤𝑒𝑟 𝑅𝑢𝑙𝑒]
3.
𝑑
𝑑𝑥(𝑎𝑥 +𝑏)𝑛)=𝑎𝑛(𝑎𝑥 + 𝑏)𝑛−1 [𝐶𝑜𝑚𝑝𝑜𝑠𝑖𝑡𝑒 𝑅𝑢𝑙𝑒]
4.
𝑑
𝑑𝑥(𝑓(𝑥)± 𝑔(𝑥))= 𝑓′(𝑥) ±𝑔′(𝑥)
5.
𝑑
𝑑𝑥(𝑢𝑣)= 𝑢 𝑑𝑣
𝑑𝑥+𝑣𝑑𝑢
𝑑𝑥 [𝑃𝑟𝑜𝑑𝑢𝑐𝑡 𝑅𝑢𝑙𝑒]
6.
𝑑
𝑑𝑥(𝑢
𝑣) = 𝑣𝑑𝑢
𝑑𝑥 −𝑢𝑑𝑣
𝑑𝑥
𝑣2 [𝑄𝑢𝑜𝑡𝑖𝑒𝑛𝑡 𝑅𝑢𝑙𝑒]
7.
𝑑𝑦
𝑑𝑥 =𝑑𝑦
𝑑𝑢× 𝑑𝑢
𝑑𝑥 [𝐶ℎ𝑎𝑖𝑛 𝑅𝑢𝑙𝑒]
8.
𝑑
𝑑𝑥(𝑒𝑥)= 𝑒𝑥
9.
𝑑
𝑑𝑥(𝑒𝑎𝑥+𝑏)= 𝑒𝑎𝑥+𝑏 ×𝑑
𝑑𝑥(𝑎𝑥 +𝑏)
10.
𝑑
𝑑𝑥(𝑙𝑛 |𝑥|) =1
𝑥
11.
𝑑
𝑑𝑥[ln |𝑎𝑥+𝑏|]=1
𝑎𝑥+𝑏 ×𝑑
𝑑𝑥(𝑎𝑥 +𝑏)
12.
𝑑
𝑑𝑥(sin𝑥)= cos𝑥
13.
𝑑
𝑑𝑥(cos𝑥)= − sin𝑥
