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Derivatives and limits of Functions
Typology: Lecture notes
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Differential Calculus 1 (Lecture)
EMAT 0103 BSCE 1D | Ma’am Maglaque | SEM 1 2022
Interval notation is a way of writing
solutions to algebraic inequalities.
Unbounded Intervals
Compound Inequality
Bounded Intervals
If
y=f ( x )=
5 x+ 3
4 x− 5
, show that x=f ( y )
Show that
f ( x+ 3 )
f ( x − 1 )
=f ( 4 ) , if f
x
x
INTERVALS FUNCTIONS
Differential Calculus 1 (Lecture)
EMAT 0103 BSCE 1D | Ma’am Maglaque | SEM 1 2022
n
n
n
Differential Calculus 1 (Lecture)
EMAT 0103 BSCE 1D | Ma’am Maglaque | SEM 1 2022
THEOREMS ON LIMITS OF
FUNCTIONS
Limits of a constant
lim
x→ a
k=k
, where k is constant / any
real number
Limits of the identity function
lim
x→ a
x =a
Limits of a constant times a function
lim
x→ a
k f ( x )
=¿ k lim
x→ a
f ( x )¿
Limits of the sum/difference of two
functions
lim
x→ a
f ( x ) ± g ( x )
=lim
x → a
f ( x)± lim
x → a
g( x )
Limits of the product of two functions
lim
x→ a
x→ a
f ( x ) lim
x →a
g ( x)
Limits of the quotient of two functions
lim
x→ a
f (x)
g ( x)
lim
x → a
f (x)
lim
x→ a
g( x)
,
lim
x→ a
g ( x)≠ 0
Limits of the nth power of a function
lim
x→ a
f ( x )
n
lim
x → a
f ( x )
n
, where n is any
real number
Limits of the nth root of a function
lim
x→ a
n
f ( x ) ]=
n
lim
x→ a
f ( x )
Infinite Limits
Examples:
lim
x→
1
2
4 x
=
x → 5
x + 3
x− 1
=
lim
x→− 1
( x+ 6 )
√
x
2
Differential Calculus 1 (Lecture)
EMAT 0103 BSCE 1D | Ma’am Maglaque | SEM 1 2022
x → 2
x
2
x− 2
=
x → 3
x
3 − 27
x
2
=
lim
h→ 0
( x +h)
2
−x
2
h
=
Differential Calculus 1 (Lecture)
EMAT 0103 BSCE 1D | Ma’am Maglaque | SEM 1 2022
lim
x→ ∞
3 x
3
x
2
=
x
, compute lim
h→ 0
f ( x +h )−f (x)
h
f ( x )=
x
2
, compute lim
h→ 0
f ( x +h )−f (x)
h
Differential Calculus 1 (Lecture)
EMAT 0103 BSCE 1D | Ma’am Maglaque | SEM 1 2022
1 − 3 x , compute lim
h→ 0
f ( x +h )−f (x)
h
Differential Calculus 1 (Lecture)
EMAT 0103 BSCE 1D | Ma’am Maglaque | SEM 1 2022
4 + x
2
x
3
d (x) = derivative of x
d (y) = derivative of y
d (v) = derivative of the variable
Derivative of a constant
d (c) = 0
d (cu) =
NOTE
Differential Calculus 1 (Lecture)
EMAT 0103 BSCE 1D | Ma’am Maglaque | SEM 1 2022
y=
( 1 −x)
2
y=
1 − 3 x
2 +x
3 +x
Sum / Difference Rule
d
dx
d
dx
f ( x ) ±
d
dx
g( x )
Product Rule
If y=uv
, then
d y
dx
=u
d v
dx
+v
d u
dx
Quotient Rule
d
dx
(
f ( x)
g( x )
)
g ( x)
d
dx
f ( x )−f (x)
d
dx
g( x )
g (x)
DERIVATIVES OF ALGEBRAIC
FUNCTION
DERIVATIVE OF LOGARITHMIC
FUNCTION
Differential Calculus 1 (Lecture)
EMAT 0103 BSCE 1D | Ma’am Maglaque | SEM 1 2022
Differential Calculus 1 (Lecture)
EMAT 0103 BSCE 1D | Ma’am Maglaque | SEM 1 2022
y=
x
x
2
w=
2 y
y
2
y=
x− 4 x
1
2
y=x
3
2
1
2
y=
( 2 −x )
2
x
1
3
y=x
3
( 4 x− 1 )
4
