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.