Table of Derivatives
Throughout this table, a and b are constants, independent of x.
F 0 (x) =
F (x)
dF
dx
0
af (x) + bg(x)
f (x) + g(x)
f (x) − g(x)
af (x)
af 0 (x) + bg (x)
f 0 (x) + g 0 (x)
f 0 (x) − g 0 (x)
af 0 (x)
f (x)g(x)
f (x)g(x)h(x)
f 0 (x)g(x) + f (x)g 0 (x)
f 0 (x)g(x)h(x) + f (x)g 0 (x)h(x) + f (x)g(x)h0 (x)
f (x)
g(x)
1
g(x)
f 0 (x)g(x)−f (x)g 0 (x)
g(x)2
g 0 (x)
− g(x)
2
f g(x)
f 0 g(x) g 0 (x)
1
a
xa
g(x)a
0
0
axa−1
ag(x)a−1 g 0 (x)
sin x
sin g(x)
cos x
cos g(x)
tan x
csc x
sec x
cot x
cos x
g (x) cos g(x)
− sin x
0
−g (x) sin g(x)
sec2 x
− csc x cot x
sec x tan x
− csc2 x
ex
eg(x)
ax
ln x
ln g(x)
loga x
arcsin x
arcsin g(x)
arccos x
arctan x
arctan g(x)
arccsc x
arcsec x
arccot x
0
ex
g 0 (x)eg(x)
(ln a) ax
1
x
g 0 (x)
g(x)
1
x ln a
√ 1
1−x2
g 0 (x)
√
1−g(x)2
1
− √1−x
2
1
1+x2
g 0 (x)
1+g(x)2
1
− x√1−x
2
1
√
x 1−x2
1
− 1+x
2
Table of Indefinite Integrals
Throughout this table, a and b are given constants, independent of x
and C is an arbitrary constant.
f (x)
R
u(x)v 0 (x)
f y(x) y 0 (x)
f (x) dx
R
R
a f (x) dx + b g(x) dx + C
R
R
f (x) dx + g(x) dx + C
R
R
f (x) dx − g(x) dx + C
R
a f (x) dx + C
R
u(x)v(x) − u0 (x)v(x) dx + C
R
F y(x) where F (y) = f (y) dy
1
a
x+C
ax + C
af (x) + bg(x)
f (x) + g(x)
f (x) − g(x)
af (x)
xa
1
x
a 0
g(x) g (x)
sin x
g (x) sin g(x)
cos x
tan x
csc x
sec x
cot x
sec2 x
csc2 x
sec x tan x
csc x cot x
0
ex
eg(x) g 0 (x)
eax
ax
ln x
√ 1
1−x2
g 0 (x)
√
1−g(x)2
√ 1
a2 −x2
1
1+x2
g 0 (x)
1+g(x)2
1
a2 +x2
√1
x 1−x2
F (x) =
xa+1
a+1
+ C if a 6= −1
ln |x| + C
g(x)a+1
a+1
+ C if a 6= −1
− cos x + C
− cos g(x) + C
sin x + C
ln | sec x| + C
ln | csc x − cot x| + C
ln | sec x + tan x| + C
ln | sin x| + C
tan x + C
− cot x + C
sec x + C
− csc x + C
ex + C
eg(x) + C
1 ax
+C
a e
1
x
a +C
ln a
x ln x − x + C
arcsin x + C
arcsin g(x) + C
arcsin xa + C
arctan x + C
arctan g(x) + C
1
a
arctan xa + C
arcsec x + C
Properties of Exponentials
In the following, x and y are arbitrary real numbers, a and b are arbitrary constants that are
strictly bigger than zero and e is 2.7182818284, to ten decimal places.
1) e0 = 1, a0 = 1
2) ex+y = ex ey , ax+y = ax ay
3) e−x = e1x , a−x = a1x
y
y
4) ex = exy , ax = axy
d x
e
dx
d
= g 0 (x)eg(x) , dx
ax = (ln a) ax
R
R ax
6) ex dx = ex + C,
e dx = a1 eax + C if a 6= 0
5)
= ex ,
d g(x)
e
dx
7) lim ex = ∞, lim ex = 0
x→∞
x→−∞
x
lim a = ∞, lim ax = 0 if a > 1
x→∞
x→−∞
lim ax = 0, lim ax = ∞ if 0 < a < 1
x→∞
x→−∞
x
8) The graph of 2 is given below. The graph of ax , for any a > 1, is similar.
y
y = 2x
6
4
2
1
−3
−2
−1
1
2
x
3
Properties of Logarithms
In the following, x and y are arbitrary real numbers that are strictly bigger than 0, a is
an arbitrary constant that is strictly bigger than one and e is 2.7182818284, to ten decimal
places.
1) eln x = x, aloga x = x, loge x = ln x,
2) loga ax = x, ln ex = x
loga x =
ln x
ln a
ln 1 = 0, loga 1 = 0
ln e = 1, loga a = 1
3) ln(xy) = ln x + ln y, loga (xy) = loga x + loga y
4) ln xy = ln x − ln y, loga xy = loga x − loga y
ln y1 = − ln y, loga y1 = − loga y,
5) ln(xy ) = y ln x, loga (xy ) = y loga x
0
(x) d
1
ln(g(x)) = gg(x)
, dx loga x = x ln
a
R
R 1
7) x dx = ln |x| + C, ln x dx = x ln x − x + C
6)
d
dx
ln x = x1 ,
d
dx
8) lim ln x = ∞, lim ln x = −∞
x→∞
x→0
lim loga x = ∞, lim loga x = −∞
x→∞
x→0
9) The graph of ln x is given below. The graph of loga x, for any a > 1, is similar.
1.5
y
y = ln x
1.0
0.5
x
1
−0.5
−1.0
−1.5
2
3
4