1
D
IFFERENTIAL CALCULUS
D
IFFERENTIABILITY
LECTURE
-04
1.
Derivative:
2.
Differentiability:
3.
Differentiable Function:
A function
𝑓(?)
is said to be differentiable at
? = 𝑎
if
𝑓
′
(?)
exists at
? = 𝑎
. That is
𝐿𝑓
′
(𝑎) = ?𝑓
′
(𝑎).
4.
Left Hand Derivative (L. H. D.) and Right Hand Derivative (R. H. D.):
If
lim
ℎ→0
−
𝑓(𝑎−ℎ)−𝑓(𝑎)
−ℎ
exists then it is called the left derivative of
𝑓(?)
at
? = 𝑎
and denoted
by
𝐿𝑓
′
(𝑎).
Thus
𝐿𝑓
′
(𝑎) = lim
ℎ→0
−
𝑓(𝑎−ℎ)−𝑓(𝑎)
−ℎ
. If
lim
ℎ→0
+
𝑓(𝑎+ℎ)−𝑓(𝑎)
ℎ
exists then it is called the
right derivative of
𝑓(?)
at
? = 𝑎
and denoted by
?𝑓′(𝑎).
Thus
?𝑓
′
(𝑎) =
lim
ℎ→0
+
𝑓(𝑎+ℎ)−𝑓(𝑎)
ℎ
.

2
5.
Geometrical Significance /Interpretation of differentiation:

3
Briefly,
Let
?
= 𝑓
(?
)
be a function, and let
?
(𝑎
, 𝑓(
𝑎)
)
and
?
(𝑎
+ ℎ
, 𝑓(
𝑎
+ ℎ)
)
be two points on the graph
of the function that are close to each other. This graph is shown below:
Joining the points
?
and
?
with a straight line gives us the secant on the graph of the function. And in
the
∆???
the gradient of the line is given by
In limiting process, i.e. as
?
approaches
?
,
ℎ
becomes really small, almost close to zero. So
𝑚 =
change in y