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Derivative and Integral Reference Guide
Differentiation Rules
Linearity Product & Quotient Rules Chain Rule d dx
[
u + v
]
= u′^ + v′^ d dx
[
uv
]
= u′v + v′u d dx
[
f (u)
]
= f ′(u) · u′
d dx
[
cu
]
= cu′^ d dx
[ (^) u v
]
u′v − v′u v^2
Derivative Identities
d dx
[
c
]
d dx
[
x
]
d dx
[
un
]
= nun−^1 u′
d dx
[
eu
]
= u′eu^ d dx
[
bu
]
= ln(b)buu′
d dx
[
ln u
]
u′ u
d dx
[
logb u
]
ln b
u′ u d dx
[
sin u
]
= u′^ cos u d dx
[
cos u
]
= − u′^ sin u d dx
[
tan u
]
= u′^ sec^2 u
d dx
[
csc u
]
= − u′^ csc u cot u d dx
[
sec u
]
= u′^ sec u tan u d dx
[
cot u
]
= − csc^2 u
d dx
[
arcsin u
]
u′ √ 1 − u^2
d dx
[
arccos u
]
u′ √ 1 − u^2
d dx
[
arctan u
]
u′ 1 + u^2 d dx
[
arccsc u
]
u′ u
u^2 − 1
d dx
[
arcsec u
]
u′ u
u^2 − 1
d dx
[
arccot u
]
u′ 1 + u^2
Fundamental Theorems of Calculus
F ′(x) = f (x) =⇒
∫ (^) b
a
f (x) dx = F (b) − F (a)
d dx
[∫ (^) b(x)
a(x)
f (t) dt
]
= f
b(x)
· b′(x) − f
a(x)
· a′(x)
Source: Stewart, J. (2020). Calculus, 9e. Cengage Learning.
Integration Rules Linearity Integration by Parts ∫ (^) [ f (x) + g(x)
]
dx =
f (x) dx +
g(x) dx
u dv = uv −
v du ∫ af (x) dx = a
f (x) dx
Integral Identities ∫ 0 dx = C
dx = x + C
xn^ dx = xn+ n + 1
- C, n 6 = − 1 ∫ (^1) x dx = ln |x| + C
ex^ dx = ex^ + C
bx^ dx = bx ln b
+ C
ln x dx = x ln(x) − x + C
logb x dx =
ln(b) [x ln(x) − x] + C ∫ cos x dx = sin x + C
sin x dx = − cos x + C
sec^2 x dx = tan x + C ∫ sec x tan x dx = sec x + C
csc x cot x dx = − csc x + C
csc^2 x dx = − cot x + C ∫ tan x dx = − ln | cos x| + C
cot x dx = ln | sin x| + C ∫ sec x dx = ln | sec x + tan x| + C
csc x dx = − ln | csc x + cot x| + C ∫ √^1 a^2 − x^2
dx = arcsin x a +^ C
a^2 + x^2 dx^ =
a arctan^
x a +^ C
x
x^2 − 1
dx = arcsec x + C
Trig Sub
x = a sin θ dx = a cos θ dθ √ a^2 − x^2 = a cos θ
a (^) x
√ a^2 − x^2
θ
x = a tan θ dx = a sec^2 θ dθ √ a^2 + x^2 = a sec θ
x
a
√ a^2 + x^2
θ
x = a sec θ dx = a sec θ tan θ dθ √ x^2 − a^2 = a tan θ
x √x (^2) − a 2
a
θ
