Bright Career Science Academy
Derivatives Formulas & Rules (Edition-3)
1
1
. . . .
( ) 1 c . (x ) c (x) ( ) 0
df
g.
d f d x
P o w e r R u le w h er e P ro d u c t R u le
w h e re " " is c o n s ta n t.
Q u o ti en t R u =le • d x g
n n
n n
d d df d df dg
x n x f n f n R f g g f
dx dx d x dx dx d x
d dx d d d d c
x f f c c
dx dx dx dx dx d x
2
dg
f. d 1 df
dx R ule f o r S q u ar e R o o t f = . C h ai n R u l e
dx d x
2 f
•
g
dy dy d u
dx du d x
2
2
Derivativeof
Trigonometric Functions
d du
sinu = cosu.
dx dx
d du
cosu = si nu.
dx dx
d du
tanu = sec u.
dx dx
d du
cosec u = cosec u.cot u.
dx dx
d du
secu =s ecu.tanu.
dx dx
d du
cotu= cosec u.
dx dx
2
2
Derivativeof
HyperbolicFunctions
d du
sinhu=coshu.
dx dx
d du
coshu= sinhu.
dx dx
d du
tanhu= sech u.
dx dx
d du
cosechu = cosechu.coth u.
dx dx
d du
sechu= sechu.tanhu.
dx dx
d du
cothu= cosech u.
dx dx
1
2
1
2
1
2
1
2
1
2
1
2
Deriv ative o f
Invers e Trigon omet ric Func tions
d 1 du
sin u = .
dx dx
1 u
d 1 du
cos u = .
dx dx
1 u
d 1 du
tan u = .
dx 1+ u dx
d 1 du
cosec u = .
dx dx
u u 1
d 1 du
sec u = .
dx dx
u u 1
d 1 du
cot u = .
dx 1+ u dx
1
2
1
2
1
2
1
2
1
2
1
2
De rivati ve of
Inve rse Hy perbo lic Fu nctio ns
d 1 d u
sin h u= .
dx dx
1 + u
d 1 d u
cos h u= .
dx dx
u 1
d 1 du
tan h u= .
dx 1 u dx
d 1 du
cos ech u = .
dx dx
u 1 + u
d 1 du
sec h u= .
dx dx
u 1 u
d 1 du
cot h u= .
dx 1 u d x
u u u u
a
Derivativeof
d du d du d 1 du d 1 du
• e =e . • a =a .lna. • lnu= . • log u= .
Exponential &LogarithmicFunctions
dx dx dx dx dx u dx dx u.lna dx
n + 1
n
n+ 1
n
ax
ax
f(x )
f(x )
2 2
P o w e r R u le o f Int e g r at i o n
x
• x d x = n + 1
f
• f .f dx = n + 1
f
• dx = l n f
f
• a f( x ) d x = a f ( x ) d x
• 1 dx = x
In te g r a t io n o f Ex p o n e n t ia l F u n c t io n s
e
• e
w h e re
dx= d(a x )
dx
a
• a dx = d
ln a . f (x )
dx
dx 1
• = t
a +
n 1
w h e re n 1
x a
1
1
2 2
2 2
2 2
2
2 2 2 2 1
2
2 2 2 2 1
2
2 2 2 2 1
2 2
2 2
x
an a
dx x
• =si n a
a x
dx
• = ln x + x a
x a
x a x
• a x dx = a x + s in
2 2 a
x a x
• x + a dx = x + a + sin h
2 2 a
x a x
• x a dx = x a c o s h
2 2 a
dx 1 a + x
• = ln
a x 2a a x
dx 1 x a
• = ln
x a 2 a x + a
Integration of Trigonometric Fun ctions
cosax
• sin ax dx = d(ax)
dx
sinax
• cos ax dx= d(ax)
dx
ln sec ax ln cos ax
• t an ax dx= =
d d
(ax) (ax)
dx dx
ln cosec ax cot ax
• cosec ax dx= d(ax )
dx
ln sec ax+tan ax
• sec ax d x= d(ax)
dx
ln sin ax
• cot ax d x=
2
2
d(ax)
dx
cotax
• cosec ax dx= d(ax)
dx
tanax
• sec axdx= d(ax)
dx
cosec ax
• cosec a x.cot a x dx= d(ax)
dx
sec ax
• sec ax. tan ax dx= d(ax
:
)
“ ”
dx
.
Note Add Integration Con stant c w ith
Every Indefi nite Integration Fo rmula
Integra tionB yParts Rule
Prope rtieso fD
. . .
. (x) (x) . (x)
Prope rty-1 (
efinite Integr al
Prope rty-1is Called "Funda menta ltheor em
)
ofcalcu l s
)
u
( ) (
ax ax
b
a
d
f g dx f g dx f g dx d x
dx
e a f f dx e f
f x d x F b F a
Prope rty-2 ( ) ( )
Prop erty-3 (x) (x) (x )
Where
"
b a
a b
b c b
a a c
f x dx f x dx
f dx f dx f dx
a c b
Compiled By
: Muzzammil Subhan
M.Phil. Math
(Minhaj University )
M.Sc. Math
(Quaid-i-Azam University, Islamabad)
M.Ed.
(University of Sargodha), B.Sc., B.C.S. &
PGD-IT
Contact No: 0300-7779500
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Integration Formulas & Rules Compiled By: Muzzammil Subhan
22 2
2 2
3 3 2 2
33 3
22 2 2
A lg e b r a ic F o rm ul a s
2
3
2 2 2
a b a b ab
a b a b a b
a b a b a b a b
a b a b a b a b
a b c a b c a b b c c a
