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Differentiation Formulas

d
k=0
dx
d
[f (x) ± g(x)] = f 0 (x) ± g 0 (x)
dx
d
[k · f (x)] = k · f 0 (x)
dx
d
[f (x)g(x)] = f (x)g 0 (x) + g(x)f 0 (x)
dx


g(x)f 0 (x) − f (x)g 0 (x)
d f (x)
=
2
dx g(x)
[g(x)]
d
f (g(x)) = f 0 (g(x)) · g 0 (x)
dx
d n
x = nxn−1
dx
d
sin x = cos x
dx
d
cos x = − sin x
dx
d
tan x = sec2 x
dx
d
cot x = − csc2 x
dx
d
sec x = sec x tan x
dx
d
csc x = − csc x cot x
dx
d x
e = ex
dx
d x
a = ax ln a
dx
1
d
ln |x| =
dx
x
d
1
sin−1 x = √
dx
1 − x2
d
−1
cos−1 x = √
dx
1 − x2
d
1
tan−1 x = 2
dx
x +1
d
−1
cot−1 x = 2
dx
x +1
d
1
√
sec−1 x =
dx
|x| x2 − 1
d
−1
√
csc−1 x =
dx
|x| x2 − 1

Integration Formulas

Z
(1)

dx = x + C

(2)

Z

(3)

Z

xn dx =

xn+1
+C
n+1

(1)
(2)

dx
= ln |x| + C
x

(3)

ex dx = ex + C

(4)

(4)
Z
(5)
Z

ax dx =

(6)

1 x
a +C
ln a

(5)

Z
(7)
(8)

ln x dx = x ln x − x + C

(6)

sin x dx = − cos x + C

(7)

cos x dx = sin x + C

(8)

tan x dx = − ln | cos x| + C

(9)

Z

(9)

Z

(10)

Z

(11)
Z
cot x dx = ln | sin x| + C

(10)

sec x dx = ln | sec x + tan x| + C

(11)

(12)
Z
(13)
Z
csc x dx = − ln | csc x + cot x| + C (12)

(14)
(15)

Z

(16)

Z

sec2 x dx = tan x + C

(13)

csc2 x dx = − cot x + C

(14)

sec x tan x dx = sec x + C

(15)

csc x cot x dx = − csc x + C

(16)

x
dx
= sin−1 + C
a
a2 − x2

(17)

dx
1
x
= tan−1 + C
a2 + x2
a
a

(18)

dx
1
|x|
√
= sec−1
+C
2
2
a
a
x x −a

(19)

(17)
Z
(18)
Z
(19)
Z
(20)
Z
(21)
(22)

Z

√



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