Question

Discuss the Differentiability of f(x)=Ix - a I

Answer 1

Answer:

So the function f(x) = |x| is not differentiable

Step-by-step explanation:

domain and range

Example (continued)

When not stated we assume that the domain is the Real Numbers.

For x2 + 6x, its derivative of 2x + 6 exists for all Real Numbers.

So we are still safe: x2 + 6x is differentiable.

But what about this:

Example: The function f(x) = |x| (absolute value):

|x| looks like this:   Absolute Value function

At x=0 it has a very pointy change!

Does the derivative exist at x=0?

Testing

We can test any value "c" by finding if the limit exists:

limh→0  f(c+h) − f(c)h

Example (continued)

Let's calculate the limit for |x| at the value 0:

 

Start with:  limh→0  f(c+h) − f(c)h

f(x) = |x|:  limh→0  |c+h| − |c|h

c=0:  limh→0  |h| − |0|h

Simplify:  limh→0  |h|h

The limit does not exist! To see why, let's compare left and right side limits:

From Left Side:  limh→0−  |h|h = −1

From Right Side:  limh→0+  |h|h = +1

The limits are different on either side, so the limit does not exist.