Examining differentiability Using Graph of Function - Practice Questions & MCQ

Examining differentiability Using Differentiation and Graph

1. Using Differentiation (only for continuous functions)

There are some functions which are defined piecewise, in such cases first we need to check if the function is continuous at the split point, and if it is continuous we need to differentiate each branch function and compare left - hand and right - hand derivative at the split point.

2. Differentiability using Graphs

A function f(x) is not differentiable at x = a if

1. Function is discontinuous at x = a
2. Graph of a function has a sharp turn at x = a
3. Function has a vertical tangent at x = a

Illustration 1

Check the differentiability of the following function.

1. f(x) = sin |x|

Method 1

Using graphical transformation, we can draw its graph

Using graph we can tell that at x = 0, the graph has a sharp turn, so it is not differentiable at x = 0.

Method 2

As LHL = RHL = f(0) = 0, so the function is continuous at x = 0

So we can use differentiation to check differentiability

As these are not equal, so, f(x) = sin |x| is not differentiable at x = 0

Illustration 2

f(x) = |log |x||, x not equal to 0

Plot the graph of | log |x| | using graphical transformation

We can see that graph has sharp turn at +1 and -1 so function is not differentiable at these points.