Let f (x) be a real valued function defied on
an open interval (a, b) and let c ϵ (a, b). Then f (x) is said to be
differentiable or derivable at x = c, if exist finitely.
This limit is called the derivative or differential coefficient of the function f (x) at x = c, and is denoted f’ (c) or Df (c) or {d/dx [f(x)]}ₓ = c. Thus, f (x) is differentiable at x = c
This limit is called the derivative or differential coefficient of the function f (x) at x = c, and is denoted f’ (c) or Df (c) or {d/dx [f(x)]}ₓ = c. Thus, f (x) is differentiable at x = c
The limits is called the left
hand derivative of f (x) at x = c and is denoted by f’ (c) or, Lf’ (c), while is called the right
hand derivative of f (x) at x = c and is denoted by f’ (c⁺) or Rf’(c).
Thus, f (x) is differentiable at x = c ⇒ Lf’ (c) = Rf’(c). If Lf’ (c) ≠ Rf’(c), we say that f (x) is not differentiable
at x = c.
Meaning of differentiability of a
function at a point: f
(x) is differentiable at point P, iff there exists a unique tangent at point P.
in order words, f (x) is differentiable at a point P iff the curve does not
have P as a corner point.
Consider the function f (x) = |x|. This
function is not differentiable at x = 0, because if we draw tangent at the
origin as the limiting position of the chords on the left hand side of the
origin, it is the line y = - x whereas the tangent at the origin as the
limiting position of the chords on the right hand side of the origin is the
line y = x. mathematically, left hand derivative at the origin is - 1 (slope of
the line y = - x) and the fight hand derivate at the origin is 1 (slope of the
line y = x).
Let f (x) be a differentiable function at a
point P. then the curve y = f (x) has a unique tangent at P. since tangent at P
is the limiting position of the chord PQ when Q → P. So, if f (x) is
differentiable at a point P, then chords exist on both sides of P. consequently
f (x) is continuous at P.
It follows from the above discussion that, if
a function is not differentiable at x = c, then either it has (c, f (c)) as a
corner point or it is discontinuous at x = c.
Also, every differentiable function is
continuous.
Relation between continuity and
differentiability: In
the above discussion, we have observed that if a function is differentiable at
a point, then it should be continuous at that point and a discontinuous
function cannot be differentiable. This fact is provided in the following
theorem.
If a function I differentiable at a point, it
is necessarily continuous at that point. But the converse is not necessarily
true.
Or
f (x) is differentiable at x = c ⇒ f (x)
is continuous at x = c.
Converse: The
converse of the above theorem is not necessarily true i.e. a function may be
continuous at a point but may not be differentiable at that point because f (x)
= |x| is continuous at x = 0 but it is not differentiable at x = 0.
Evaluate: show
that f (x) = |x| is not differentiable at x = 0.
Solution: we have,
(LHD at x = 0)
∴
(LHD at x = 0) ≠ (RHD at x = 0).
So, f (x) is not differentiable at x = 0.
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