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1. The derivative of the function
f
at the point
x=a
is
f
'
( a )=lim
h → 0
f ( a+h) −f ( a )
h
2. The derivative of the function
f
is given by
f
'
x
=lim
h → ∞
f ( x+ h)−f (x )
h
3. If
f
is differentiable at
x=a
, then
f
is continuous at
x=a
Power Rule
d
dx
c= 0 , for any constant
c
d
dx
x= 1
.
d
dx
x
n
=n x
n− 1
, for any integer
n> 0
7. General Power Rule
d
dx
x
r
=r x
r− 1
, for any real number r ≠ 0.
8. If
f ( x) and
g( x ) are differentiable at
x and
c is any constant, then
i.
d
dx
[ f ( x ) + g ( x) ]=f
'
( x )+ g ' (x)
ii.
d
dx
[
f ( x )−g ( x ) ]
=f
'
( x )−g ' ( x)
iii.
d
dx
[
cf ( x ) ]
=cf ' ( x)
- Product Rule
d
dx
[
f ( x ) g ( x ) ]
=f
'
( x ) g ( x ) + f ( x ) g ' (x ) .
- Quotient Rule
d
dx
[
f ( x )
g ( x )
]
f
'
( x ) g ( x )−f ( x) g '(x )
- Chain Rule
CH:2 DIFFERENTIATION RULES
d
dx
[
f
g ( x )
]
=f
'
g ( x )
g '(x ) .
12. If
f
is differentiable on its domain and has an inverse
g=f
− 1
g
'
( x ) =
f
'
( g
x
(f ’( g( x )) ≠ 0 )
- Trigonometric Derivatives
i.
d
dx
sin x=cos x
ii.
d
dx
cos x=−sin x
iii.
d
dx
tan x=sec
2
x
iv.
d
dx
cot x=−csc
2
x
v.
d
dx
sec x=sec x tan x
d
dx
csc x=¿−csc x cot x ¿ Similarly,
i.
d
dx
sin
f ( x )
=cos
f ( x )
f ' (x)
ii.
d
dx
cos
f ( x)
=−sin
f ( x )
f ' (x)
iii.
d
dx
tan ( f ( x) ) =sec
2
( f ( x ) ) f ' (x)
iv.
d
dx
cot
f ( x )
=−csc
2
f ( x )
f ' (x )
v.
d
dx
sec
f ( x )
=sec
f ( x )
tan
f ( x )
f ' ( x)
vi.
d
dx
csc(f ( x) )=¿−csc (f ( x ))cot( f ( x) )¿
Exponential Derivatives
d
dx
a
x
=a
x
ln a
, for
a> 0
d
dx
a
f ( x)
=ln a a
f (x )
f
'
( x )
.
d
dx
e
x
=e
x
d
dx
e
f ( x)
=e
f ( x)
f
'
( x ) Logarithmic Derivatives
d
dx
d
dx
ln ( f ( x ) ) =
f
'
x
f ( x )
Implicit Differentiation
d
dx
g ( y )=g
'
( y ) y
'
( x )
- Inverse Trigonometric Derivatives
viii.
csc θ=
sin θ
ix.
sec θ=
cos θ
x.
cot θ=
tan θ
xi.
sin(α ± β)=sin α cos β ± cos α sin β
xii.
cos (α ± β)=cos α cos β ± sin α sin β
