Common Derivatives
𝑑
𝑥 =1
𝑑𝑥
𝑑
𝑑
[𝑎𝑓 𝑥 ] = 𝑎 [𝑓 𝑥 ]
𝑑𝑥
𝑑𝑥
Limits & Derivatives Cheat Sheet
𝑑
𝑎𝑥 = 𝑎
𝑑𝑥
𝑑
𝑎𝑥 𝑛 = 𝑛𝑎𝑥 𝑛−1
𝑑𝑥
Properties of Limits
𝑑
𝑐 =0
𝑑𝑥
𝑑
[𝑓 𝑥 ]𝑛 = 𝑛[𝑓 𝑥 ]𝑛−1 𝑓′(𝑥)
𝑑𝑥
lim 𝑐𝑓 𝑥
= 𝑐 lim 𝑓(𝑥)
𝑥→𝑎
𝑥→𝑎
𝑑 1
= −𝑛𝑥 −
𝑑𝑥 𝑥 𝑛
lim [𝑓 𝑥 ± 𝑔 𝑥 ] = lim 𝑓(𝑥) ± lim 𝑔(𝑥)
𝑥→𝑎
𝑥→𝑎
𝑥→𝑎
lim [𝑓 𝑥 𝑔 𝑥 ] = lim 𝑓(𝑥) lim 𝑔(𝑥)
𝑥→𝑎
lim
𝑥→𝑎
𝑥→𝑎
𝑥→𝑎
lim [𝑓 𝑥 ]𝑛 = lim 𝑓 𝑥
𝑥→𝑎
𝑛
𝑥→𝑎
Limit Evaluations At ±∞
lim 𝑒 𝑥 = lim 𝑒 𝑥 = 0
𝑥→+∞
𝑥→−∞
lim ln 𝑥 = ∞ and lim+ ln 𝑥 = −∞
𝑥→∞
𝑥→0
𝑐
if r > 0: lim 𝑟 = 0
𝑥→∞ 𝑥
𝑐
if r > 0 & ∀𝑥 > 0 𝑥 𝑟 ∈ ℝ ∶ lim 𝑟 = 0
𝑥→−∞ 𝑥
lim 𝑥 𝑟 = ∞ for even r
𝑥→±∞
lim 𝑥 𝑟 = ∞ and lim 𝑥 𝑟 = −∞ for odd r
𝑥→+∞
𝑥→−∞
L’Hopital’s Rule
𝑓(𝑥) 0
±∞
𝑓(𝑥)
𝑓′(𝑥)
If lim
= or
then lim
= lim
𝑥→𝑎 𝑔(𝑥)
𝑥→𝑎 𝑔(𝑥)
𝑥→𝑎 𝑔′(𝑥)
0
±∞
Derivative Definition
𝑑
𝑓 𝑥
𝑑𝑥
= 𝑓 ′ 𝑥 = lim
ℎ→0
𝑓 𝑥 + ℎ − 𝑓(𝑥)
ℎ
Product Rule
𝑓 𝑥 𝑔 𝑥
′
= 𝑓 ′ 𝑥 𝑔 𝑥 + 𝑓 𝑥 𝑔′(𝑥)
Quotient Rule
𝑑 𝑓 𝑥
𝑑𝑥 𝑔 𝑥
=
𝑓 ′ 𝑥 𝑔 𝑥 − 𝑓 𝑥 𝑔′(𝑥)
[𝑔 𝑥 ]2
Chain Rule
𝑑
𝑓 𝑔 𝑥
𝑑𝑥
=
𝑓′
𝑔 𝑥 𝑔′(𝑥)
Basic Properties of Derivatives
𝑐𝑓 𝑥
𝑓 𝑥 ±𝑔 𝑥
′
= 𝑐[𝑓 ′ 𝑥 ]
′
= 𝑓′(𝑥) ± 𝑔′(𝑥)
=−
𝑛
𝑥 𝑛+1
Derivatives of Trigonometric Functions
𝑥→𝑎
lim 𝑓(𝑥)
= 𝑥→𝑎
𝑖𝑓 lim 𝑔(𝑥) ≠ 0
𝑥→𝑎
lim 𝑔(𝑥)
𝑓 𝑥
𝑔 𝑥
𝑛+1
𝑑
sin 𝑥 = cos 𝑥
𝑑𝑥
𝑑
cos 𝑥 = − sin 𝑥
𝑑𝑥
𝑑
tan 𝑥 = sec 2 𝑥
𝑑𝑥
𝑑
sec 𝑥 = sec 𝑥 tan 𝑥
𝑑𝑥
𝑑
csc 𝑥 = − csc 𝑥 cot 𝑥
𝑑𝑥
𝑑
cot 𝑥 = −csc 2 𝑥
𝑑𝑥
Derivatives of Exponential & Logarithmic Functions
𝑑 𝑥
𝑒 =𝑥
𝑑𝑥
𝑑 𝑥
𝑎 = 𝑎 𝑥 ln 𝑎
𝑑𝑥
𝑑
1
ln |𝑥| =
𝑑𝑥
𝑥
𝑑
1
ln 𝑥 = , 𝑥 > 0
𝑑𝑥
𝑥
𝑑
1
log𝑎 𝑥 =
𝑑𝑥
𝑥 ln 𝑎
𝑑 𝑓(𝑥)
𝑒
= 𝑓′(𝑥)𝑒 𝑓(𝑥)
𝑑𝑥
𝑑
𝑓 𝑥
𝑑𝑥
𝑔 𝑥
=𝑓 𝑥
𝑔 𝑥
𝑑
𝑓′(𝑥)
ln 𝑓(𝑥) =
𝑑𝑥
𝑓(𝑥)
𝑑 𝑓
𝑎
𝑑𝑥
𝑥
= 𝑎 𝑓(𝑥) ln 𝑎 𝑓′(𝑥)
𝑔 𝑥 𝑓′ 𝑥
+ ln 𝑓 𝑥
𝑓 𝑥
𝑔′ 𝑥
Derivatives of Inverse Trig Functions
𝑑
1
sin−1 𝑥 =
𝑑𝑥
1 − 𝑥2
𝑑
1
sec −1 𝑥 =
𝑑𝑥
|𝑥| 𝑥 2 − 1
𝑑
1
cos−1 𝑥 = −
𝑑𝑥
1 − 𝑥2
𝑑
1
csc −1 𝑥 = −
𝑑𝑥
|𝑥| 𝑥 2 − 1
𝑑
1
tan−1 𝑥 =
𝑑𝑥
1 + 𝑥2
𝑑
1
cot−1 𝑥 = −
𝑑𝑥
1 + 𝑥2
Derivatives of Hyperbolic Functions
𝑑
sinh 𝑥 = cosh 𝑥
𝑑𝑥
𝑑
sech 𝑥 = − coth 𝑥 csch 𝑥
𝑑𝑥
𝑑
cosh 𝑥 = sinh 𝑥
𝑑𝑥
𝑑
csch 𝑥 = − tanh 𝑥 sech 𝑥
𝑑𝑥
𝑑
tanh 𝑥 = 1 − tanh2 𝑥
𝑑𝑥
𝑑
coth 𝑥 = −1 − coth2 𝑥
𝑑𝑥
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Integrals of Trigonometric Functions
න cos 𝑥 𝑑𝑥 = sin 𝑥 + 𝐶
න csc 𝑥 cot 𝑥 𝑑𝑥 = − csc 𝑥 + 𝐶
න sin 𝑥 𝑑𝑥 = −cos 𝑥 + 𝐶
න sec 𝑥 tan 𝑥 𝑑𝑥 = sec 𝑥 + 𝐶
න sec 2 𝑥 𝑑𝑥 = tan 𝑥 + 𝐶
න csc 2 𝑥 𝑑𝑥 = − cot 𝑥 + 𝐶
න cot 𝑥 𝑑𝑥 = ln | sin 𝑥 | + 𝐶
න csc 𝑥 𝑑𝑥 = ln | csc 𝑥 − cot 𝑥 | + 𝐶
න sinh 𝑥 𝑑𝑥 = cosh 𝑥 + 𝐶
න cosh 𝑥 𝑑𝑥 = sinh 𝑥 + 𝐶
Integration Cheat Sheet
Integration Properties
𝑏
𝑏
න 𝑐𝑓 𝑥 𝑑𝑥 = 𝑐 න 𝑓 𝑥 𝑑𝑥
𝑎
𝑎
𝑏
𝑏
𝑏
න 𝑓 𝑥 ± 𝑔 𝑥 𝑑𝑥 = න 𝑓(𝑥) 𝑑𝑥 ± න 𝑔(𝑥) 𝑑𝑥
𝑎
𝑎
𝑎
𝑎
න 𝑓(𝑥) 𝑑𝑥 = 0
𝑎
𝑏
𝑎
න 𝑓(𝑥) 𝑑𝑥 = − න 𝑓(𝑥) 𝑑𝑥
𝑎
𝑏
𝑏
𝑐
න tan 𝑥 𝑑𝑥 = ln | sec 𝑥 | + 𝐶
𝑐
න 𝑓(𝑥) 𝑑𝑥 + න 𝑓(𝑥) 𝑑𝑥 = න 𝑓 𝑥 𝑑𝑥
𝑎
𝑏
𝑎
Fundamental Theorem of Calculus
න sec 𝑥 𝑑𝑥 = ln | sec 𝑥 + tan 𝑥 | + 𝐶
𝑏
න 𝑓(𝑥) 𝑑𝑥 = [𝐹 𝑥 ]𝑏𝑎 = 𝐹 𝑏 − 𝐹 𝑎
𝑎
න
where 𝑓 is continuous on 𝑎, 𝑏 and 𝑓 ′ = 𝐹
Definite Integral Definition
න
1
1
𝑥
𝑑𝑥 = tan−1
+𝐶
𝑎2 + 𝑥 2
𝑎
𝑎
1
𝑎2
−
𝑥2
𝑑𝑥 = sin−1
𝑥
+𝐶
𝑎
𝑛
න 𝑓(𝑥) 𝑑𝑥 = lim ෍ 𝑓 𝑥𝑘 ∆𝑥
𝑛→∞
𝑎
where ∆𝑥 =
Partial Fractions
𝑘=1
𝑏−𝑎
𝑎𝑛𝑑 𝑥𝑘 = 𝑎 + 𝑘∆𝑥
𝑛
Common Integrals
න 𝑘 𝑑𝑥 = 𝑘𝑥 + 𝐶
1
න 𝑑𝑥 = ln |𝑥| + 𝐶
𝑥
න 𝑒 𝑥 𝑑𝑥 = 𝑒 𝑥 + 𝐶
න
න ln 𝑥 𝑑𝑥 = 𝑥 ln 𝑥 − 𝑥 + 𝐶
න 𝑏 𝑥 𝑑𝑥 =
𝑏𝑥
+𝐶
ln 𝑏
1
1
𝑑𝑥 = ln |𝑎𝑥 + 𝑏| + 𝐶
𝑎𝑥 + 𝑏
𝑎
න
න
𝑥 𝑛+1
න 𝑥 𝑛 𝑑𝑥 =
+𝐶
𝑛+1
1
1
𝑥−𝑎
𝑑𝑥 =
ln
2
2
𝑥 −𝑎
2𝑎
𝑥+𝑎
1
𝑥 2 ± 𝑎2
𝑅(𝑥)
𝐴1
𝐴2
𝐴𝑛
=
+
+ ⋯+
,
𝑄(𝑥)
𝑎1 𝑥 + 𝑏1
𝑎2 𝑥 + 𝑏2
(𝑎𝑛 𝑥 + 𝑏𝑛 )
where 𝑄 𝑥 = 𝑎1 𝑥 + 𝑏1 𝑎2 𝑥 + 𝑏2 … (𝑎𝑛 𝑥 + 𝑏𝑛 )
𝑅(𝑥)
𝐴1
𝐴2
𝐴𝑛
=
+
+⋯+
2
𝑄(𝑥) 𝑎1 𝑥 + 𝑏1 (𝑎1 𝑥 + 𝑏1 )
𝑎1 𝑥 + 𝑏1
where a linear factor of 𝑄 𝑥 is repeated 𝑛 times
𝑛
𝑅(𝑥)
𝐴𝑥 + 𝐵
= 2
𝑄(𝑥) 𝑎𝑥 + 𝑏𝑥 + 𝑐
where 𝑄 𝑥 has a factor 𝑎𝑥 2 + 𝑏𝑥 + 𝑐, where 𝑏 2 − 4𝑎𝑐 < 0
𝑅(𝑥)
𝐴1 𝑥 + 𝐵1
𝐴2 𝑥 + 𝐵2
𝐴𝑛 𝑥 + 𝐵𝑛
= 2
+
+ ⋯+
𝑄 𝑥
𝑎𝑥 + 𝑏𝑥 + 𝑐 (𝑎𝑥 2 + 𝑏𝑥 + 𝑐)2
𝑎𝑥 2 + 𝑏𝑥 + 𝑐
where 𝑄 𝑥 has a factor
𝑎𝑥 2
+ 𝑏𝑥 + 𝑐 , where
𝑏2
− 4𝑎𝑐 < 0
Integration by Parts
න 𝑢 𝑑𝑣 = 𝑢𝑣 − න 𝑣 𝑑𝑢,
𝑛
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𝑏
where 𝑣 = න 𝑑𝑣
or න 𝑓 𝑥 𝑔′(𝑥) 𝑑𝑥 = 𝑓 𝑥 𝑔 𝑥 − න 𝑓 ′ 𝑥 𝑔(𝑥) 𝑑𝑥
𝑑𝑥 = ln 𝑥 + 𝑥 2 ± 𝑎2
Integrals of Symmetric Functions
Integration by Substitution
𝑏
𝑔(𝑏)
න 𝑓 𝑔 𝑥 𝑔′(𝑥) 𝑑𝑥 = න
𝑎
𝑔(𝑎)
𝑓 𝑢 𝑑𝑢
where 𝑢 = 𝑔 𝑥 and 𝑑𝑢 = 𝑔′ 𝑥 𝑑𝑥
𝑎
𝑎
If 𝑓 is even 𝑓 −𝑥 = 𝑓 𝑥 , then න 𝑓(𝑥) 𝑑𝑥 = 2 න 𝑓 𝑥 𝑑𝑥
−𝑎
0
𝑎
If 𝑓 is odd 𝑓 −𝑥 = −𝑓 𝑥 , then න 𝑓(𝑥) 𝑑𝑥 = 0
−𝑎
Applications of Integration
𝑏
Area Between 2 Curves
Integration & Multivariate Calculus
Cheat Sheet
𝐴 = න |𝑓 𝑥 − 𝑔 𝑥 | 𝑑𝑥
𝑎
𝑛
𝑏
lim ෍ 𝐴(𝑥𝑖∗ )∆𝑥 = න 𝐴(𝑥) 𝑑𝑥
Volume Definition
𝑛→∞
𝑎
𝑖=1
Integration by Trigonometric Substitution
Expression
Substitution
Evaluation
Identity Used
𝑎2 − 𝑥 2
𝑥 = 𝑎 sin 𝜃
𝑑𝑥 = 𝑎 cos 𝜃 𝑑𝜃
𝑎2 − 𝑎2 sin2 𝜃 = 𝑎 cos 𝜃
1 − sin2 𝜃 = cos 2 𝜃
𝑥 2 − 𝑎2
𝑥 = 𝑎 sec 𝜃
𝑑𝑥 = 𝑎 sec 𝜃 tan 𝜃 𝑑𝜃
𝑎2 sec 2 𝜃 − 𝑎2 = 𝑎 tan 𝜃
sec2 𝜃 − 1 = tan2 𝜃
𝑎2 + 𝑥 2
𝑥 = 𝑎 tan 𝜃
𝑑𝑥 = 𝑎 sec2 𝜃 𝑑𝜃
𝑎2 + 𝑎2 tan2 𝜃 = 𝑎 sec 𝜃
1 + tan2 𝜃 = sec 2 𝜃
Strategy for Evaluating ‫𝒙𝒅 𝒙 𝒏𝐬𝐨𝐜 𝒙 𝒎𝐧𝐢𝐬 ׬‬
Strategy for Evaluating ‫𝒙𝒅 𝒙 𝒏 𝐜𝐞𝐬 𝒙 𝒎𝐧𝐚𝐭 ׬‬
If 𝑛 is odd
If 𝑛 is even
න sin𝑚 𝑥 cos 2𝑘+1 𝑥 𝑑𝑥 = න sin𝑚 𝑥 cos2 𝑥
= න sin𝑚 𝑥 1 − sin2 𝑥
𝑘
𝑘 cos 𝑥 𝑑𝑥
න tan𝑚 𝑥 sec 2𝑘 𝑥 𝑑𝑥 = න tan𝑚 𝑥 sec 2 𝑥
= න tan𝑚 𝑥 1 + tan2 𝑥
cos 𝑥 𝑑𝑥
Then substitute: 𝑢 = sin 𝑥
න sin2𝑘+1 𝑥 cos 𝑛 𝑥 𝑑𝑥 = න sin2 𝑥
𝑘
If 𝑚 is odd
𝑘 cos 𝑛 𝑥 sin 𝑥 𝑑𝑥
න tan2𝑘+1 𝑥 sec 𝑛 𝑥 𝑑𝑥
cos 𝑛 𝑥 sin 𝑥 𝑑𝑥
= න tan2 𝑥
Then substitute: 𝑢 = cos 𝑥
𝑑
𝑓 𝑡 𝐮𝑡
𝑑𝑡
𝑑
𝐮 𝑡 ∙𝐯 𝑡
𝑑𝑡
𝑑
𝐮 𝑡 ×𝐯 𝑡
𝑑𝑡
𝑑
𝐮 𝑓 𝑡
𝑑𝑡
sec 𝑛−1 𝑥 sec 𝑥 tan 𝑥 𝑑𝑥
Then substitute: 𝑢 = sec 𝑥
Product Identities
න sin 𝑚𝑥 cos 𝑛𝑥 𝑑𝑥
sin 𝐴 cos 𝐵
1
= sin 𝐴 − 𝐵 + sin 𝐴 + 𝐵
2
න sin 𝑚𝑥 sin 𝑛𝑥 𝑑𝑥
sin 𝐴 sin 𝐵
1
= cos 𝑎 − 𝐵 − cos 𝐴 + 𝐵
2
න cos 𝑚𝑥 cos 𝑛𝑥 𝑑𝑥
cos 𝐴 cos 𝐵
1
= cos 𝐴 − 𝐵 + cos 𝐴 + 𝐵
2
Derivatives of Vector Functions
𝑑
𝑐𝐮 𝑡
𝑑𝑡
𝑘
= න sec 2 𝑥 − 1 sec 𝑛−1 𝑥 sec 𝑥 tan 𝑥 𝑑𝑥
If 𝑛 and 𝑚 are even
Use the half angle identities:
1
1
sin2 𝑥 = 1 − cos 2𝑥 and cos2 𝑥 = 1 + cos 2𝑥
2
2
1
or this identity: sin 𝑥 cos 𝑥 = sin 2𝑥
2
𝑑
𝐮 𝑡 +𝐯 𝑡
𝑑𝑡
sec 2 𝑥 𝑑𝑥
Then substitute: 𝑢 = tan 𝑥
If 𝑚 is odd
= න 1 − cos2 𝑥
𝑘−1
𝑘−1 sec 2 𝑥 𝑑𝑥
= 𝐮′ 𝑡 + 𝐯 ′ 𝑡
= 𝑐𝐮′(𝑡)
= 𝑓 ′ 𝑡 𝐮 𝑡 + 𝑓 𝑡 𝐮′(𝑡)
= 𝐮′ 𝑡 ∙ 𝐯 𝑡 + 𝐮(𝑡) ∙ 𝐯′(𝑡)
Definite Integral of a Vector Function
𝑏
න 𝐫(𝑡) 𝑑𝑡
= 𝐮′ 𝑡 × 𝐯 𝑡 + 𝐮(𝑡) × 𝐯′(𝑡)
= 𝑓 ′ 𝑡 𝐮′ 𝑓 𝑡 ,
𝑎
𝑏
𝑏
𝑏
= න 𝑓(𝑡) 𝑑𝑡 𝐢 + න 𝑔(𝑡) 𝑑𝑡 𝐣 + න ℎ(𝑡) 𝑑𝑡 𝐤
(chain rule)
𝑎
𝑎
𝑎
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