Hey there! We receieved your request
Stay Tuned as we are going to contact you within 1 Hour
One of our academic counsellors will contact you within 1 working day.
Click to Chat
1800-5470-145
+91 7353221155
Use Coupon: CART20 and get 20% off on all online Study Material
Complete Your Registration (Step 2 of 2 )
Sit and relax as our customer representative will contact you within 1 business day
OTP to be sent to Change
Dear student,
We will learn about differentiability , and the various factors affecting differentiability. We have already defined the derivative of a function f(x) at a particular point 'a' and derivative of f(x) in general for the variable x as f(a) and f(x) respectively. The restriction in both the cases is that 'the limit must exist'.
If
does not exist, then we say that the function is not differentiable.
If the above limit exists, we say the function f(x) is differentiable.
In order to test the differentiability of a function at a point, the right hand derivative and left hand derivatives are introduced as follows:
Let f be a function of x (y=f(x)). Let a be a point in the domain of f. The RHD of f at a is defined as
where h>0, provided the limit exists.
The LHD of f at a is defined as
In the above definition showiing differentiabilty, substitute a + h = x, then h = x - a as
Rf '(a) can be rewritten as
Similarly, substitute a - h = x. Lf '(a) can be written as
Diffferentiability at a point 'a' for a function f(x) if
(i) both Rf '(a) and Lf '(a) exists and finite.
(ii) Rf '(a) = Lf '(a)
Consider the function y=|x|. This function is differentiable on (,0) and (0,), but not differentiable at x = 0.
For x>0, we have
Since the limit exists, f(x) is differentiable at x>0. Similarly, we can show that f(x) is differentiable at x<0.
We shall find RHD and LHD of f(x) at x = 0.
= -1
Therefore y = |x| is not differentiable at x = 0.
Register Yourself for a FREE Demo Class by Top IITians & Medical Experts Today !