Concept of Differentiability and Algebra of derivatives :

Differentiability:

A function is differentiable at a point when there's a defined derivative at that point. This means that the slope of the tangent line of the points from the left is approaching the same value as the slope of the tangent of the points from the right.

 The computation of the slope of a tangent line, the instantaneous rate of change of a function, and the instantaneous velocity of an object at x=a  all required us to compute the following limit.

A function f : U -> R{displaystyle f:U o mathbb {R} }, defined on an open set UÍR, is said to be differentiable at aÎU{displaystyle Usubset mathbb {R} }, {displaystyle ain U} if the derivative

exists. This implies that the function is continuous at a.

This function f is said to be differentiable on U if it is differentiable at every point of U. In this case, the derivative of f is thus a function from U into {displaystyle mathbb {R} .}R.

A continuous function is not necessarily differentiable, but a differentiable function is necessarily continuous  (at every point where it is differentiable) 

The derivative of f(x) with respect to x is the function f′(x) and is defined as,

Derivative of f at c is denoted by f ¢(c) or 

wherever the limit exists is defined to be the derivative of f.

The derivative of f is denoted by f '(x) or 

The process of finding derivative of a function is called differentiation.

Algebra of derivatives:

Properties: Let u=u(x), v=v(x) and w=w(x) be three differentiable function.

(1) (u ± v)¢ = u¢ ± v (Sum Rule of Differentiation)

(2) (uv)¢ = u¢v + uv¢ (Leibnitz or product rule)

(3),wherever v ¹ 0 (Quotient rule).

(4)  (uvw)’ = uv w’ + uwv’ + vw u’

Theorem : If a function f is differentiable at a point c, then it is also continuous at that

point.

Solution :

Since f is differentiable at c, we have

Corollary : Every differentiable function is continuous but vice versa is not true.

we have seen that the function defined by f (x) = | x| is a continuous function.

Consider the

Since the above left and right hand Derivative at 0 are not equal,

Limit does not exist and hence f is not differentiable at 0.

Thus every continuous function is not  differentiable function.