# Differentiability of the function at a point

Jun 12, 2020, 16:45 IST

## Differentiability of the function at a point

**Definition :-**Slope of the tangent at point P, which is limiting position of the chords drwn from point p and There is a unique tangent at point p

Thus, f(x) is differentiable at point p, if there exists a unique tangent at point p.

In other words, f(x) is differentiable at a ppoint p if the curve does not have p as a corner does is not differentiable at those points on which function has jumps (or holes) and sharp edges.

**Mathematically **

Ifexists then function is said to be differentiable at x= a.

is known as Right Hand Derivative (RHD)

If LHD = RHD = a finite number, then function is said to be differentiable at x = a.

**Properties of differentiability of a function : **

1. All polynomial, exponential, trigonometric, logarithmic, rational functions are differentiable in their domain.

2. Combination of two differentiable functions is a differentiable function.

3. Addition and Subtraction of one differentiable and one non-differentiable function is non-differentiable.

4. Modulus can be non-differentiable at points where expression inside modulus is zero.

5. Differentiability ⟹ Continuity

6. Discontinuity ⟹ Non-differentiability

**Example 1 :-**Test the differentiability of the function f(x) = |x - 2| at x = 2.

**Solution :-** We know that this function is continuous at x = 2.

Since the one sided derivativesdoes not exist. That is, f is not differentiable at x = 2. At all other points, the function is differentiable.

If x0 ≠ 2 is any other point then

The fact that f ′ (2) does not exist is reflected geometrically in the fact that the curve y = |x - 2| does not have a tangent line at (2, 0). Note that the curve has a sharp edge at (2, 0).

**Example 2 :-**Let f (x ) = Clearly, there is no hole (or break) in the graph of this function and hence it is continuous at all points of its domain.

**Solution :-**

Let us check whether f ′(0) exists.

Therefore, the function is not differentiable at x = 0. From the fig. 10.19, further we conclude that the tangent line is vertical at x = 0. So f is not differentiable at x = 0.

Do solve NCERT text book with the help of Entrancei **NCERT solutions for class 12 Maths.**

#### Related Link

- Number of function from set a to set b
- Inverse of matrix
- Logarithmic differentiation
- The Area of a triangle using determinant
- Differentiation of determinant
- Continuity of the function
- Differentiability of the function at a Point
- Equation of normal to the curve at a given point
- Differentiation by chain rule
- Equation of tangent line to a curve at a given point
- Area bounded by the curve
- F u and v be two functions of x, then the integral of product of these two functions is given by:
- If A and B are two finite set then the number of elements in either A or in B is given by
- If A, B and C are three finite set then the number of elements in either set A or B or in C is given by
- If set A has p no. of elements and set B has q number of elements then the total number of relations defined from set A to set B is 2pq.
- If in a circle of radius r arc length of l subtend θ radian angle at centre then
- Conversion of radian to degree and vice versa
- Addition rule of counting
- Multiplication rule of counting
- Permutation of objects
- Permutation of n object has some of repeated kind.
- Combination of objects
- Circular permutation
- Binomial Theorem
- General term of arithmatic progression
- Sum to n terms of arithmatic progression
- Insertion of n arithmetic mean in given two numbers
- Insertion of n geometric mean
- Distance formula
- Section formula
- Angle between two lines
- centroid of the triangle
- Classical probability
- Addition law probability