Question

anyone can give me maths formulas for differentiability

Answer 1

Definition of Differentiability
Differentiability at a Point

To know about differentiability at a point, first we have to know about the left hand side limit and right hand side limit.

Left Hand Limit or Derivative: 

The LHD of f at a is defined as

where, h > 0, provided the limit exists.

In the above definition showing differentiability, 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

The most important use of limit is to find the derivative of any function. 
If y = f(x) is a function and limδx→0limδx→0f(x+δx)−f(x)δxf(x+δx)−f(x)δx is the derivative of the given function at any point (x,f(x)). Then we can say that function f is differentiable at the point x if limits exists. If it does not exist, then we say that the function is not differentiable.
Right Left Hand Limit or Derivative:

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.

So, we can say that diffferentiability at a point 'a' for a function f(x) if

both Rf '(a) and Lf '(a) exists and finite.Rf '(a) = Lf '(a)

Consider the function y = |x|. This function is differentiable on (−∞−∞,0) and (0,\infy\infy), 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.

Continuity and DifferentiabilityBack to Top

If a function is differentiable at any point x, then that function is continuous at that point x. The following theorem says 'Differentiability implies continuity".

If a function f is differentiable at x = a, then it is continuous at x = a.

f is differentiable at x = a.

Now, we have

f(x) is continuous at x = a.

Every differential function is continuous, but every continuous function is not differential.

We have already shown that y = |x| is not differentiable at x = 0. It can be easily seen that 
y = |x| is continuous at x = 0.

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Answer 2

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