Derivative and Integral Reference Guide
Differentiation Rules
Linearity Product & Quotient Rules Chain Rule
0d
0
u+v=u +v
0
0 d
0
0
dx uv = u v + v u dx f(u) = f (u) · u
d
dx
0d hu i
d
dx
cu = cu
dx
0
0
v =u v − v u
v2
Derivative Identities
n
d
d
n−1 0
c = 0 dx x = 1 dx u = nu u
d
dx
u
ud
e = u0e
d
dx
d
dx
u
u 0
dx b = ln(b)b u
1
0 d
0
ln u =u u dx logb u = ln b·u u
d
d
0
0
0
2
sin u = u cos u dx cos u = − u sin u dx tan u = u sec u
d
dx
d
dx
d
d
0
0
2
csc u = − u csc u cot u dx sec u = u sec u tan u dx cot u = − csc u
0
arcsin u =u
2d
√
1 − u dx arccos u = −u0
2d
√
1 − u dx arctan u =u0
d
dx
1 + u2
0
arccsc u = −u
√ 2
d
0
u u − 1 dx arcsec u =u
d
dx
√ 2
d
0
u u − 1 dx arccot u = −u
1 + u2
Fundamental Theorems of Calculus
Zb
(
x
)
=
⇒
F
0
(
x
)
=
f
a
f(x) dx = F(b) − F(a)
dx
"
Z b(x) a(x)
d
b0(x) − fa(x)
f(t) dt
= fb(x) ·
#
· a0(x)
Integration Rules
Linearity Integration by Parts
Z
Z
Z
f(x) + g(x) dx = f(x) dx + g(x) dx
Z
Z
u dv = uv − v du
Z
af(x) dx = a
Z
f(x) dx
Z
0 dx = C
Z
1
ZZ
Integral Identities Z
dx = x + C
Z
exdx = ex + C
xdx = ln |x| + C
Z
ln x dx = x ln(x) − x + C
Z
cos x dx = sin x + C
n + 1+ C, n 6= −1
n
n+1
x dx =x
bxdx =bx
Z
sec x dx = ln |sec x + tan ln(b)[x ln(x) − x] + C
Z
x| + C 1
sin x dx = − cos x + C
Z
ZZZZZ
sec x tan x dx = sec x + C
Z
logb x dx =1
tan x dx = − ln | cos x| +
C
2
√ 2
x
Z
a − x dx = arcsin a+ C 1
ln b+ C
csc x dx = − ln | csc x +
cot x| + C
Z
2
csc x cot x dx = − csc x + sec x dx = tan x + C
Z
2
C cot x dx = ln |sin x| + C csc x dx = − cot x + C
2
1
x
Z
a2 + x dx = aarctan a+ C 1
√ 2
x x − 1dx = arcsec x + C
Trig Sub
a
x
θ
√ 2
2
√
dx = a cos θ dθ a2 − x2 a − x
x = a sin θ
= a cos θ
x = a tan θ
dx = a sec2θ dθ
θ
√ 2
a + x2 = a sec
a
θ
x
√ 2
2
a +x
√
dθ x2 − a2 = a tan
θ
√
x x2 − a2
x = a sec θ
dx = a sec θ tan θ
Source: Stewart, J. (2020). Calculus, 9e. Cengage Learning.
θ
a