Differentiable

Differentiable means that the derivative exists ...

Example: is x2 + 6x differentiable?

Derivative rules tell us the derivative of x2 is 2x and the derivative of x is 1, so:

Its derivative is 2x + 6

So yes! x2 + 6x is differentiable.

... and it must exist for every value in the function's domain.

Domain

In its simplest form the domain is
all the values that go into a function

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:

lim h→0 f(c+h) − f(c) h

Example (continued)

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

 

Start with:lim h→0 f(c+h) − f(c) h
f(x) = |x|:lim h→0 |c+h| − |c| h
c=0:lim h→0 |h| − |0| h
Simplify:lim h→0 |h| h

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

From Left Side:lim h→0 |h| h = −1
From Right Side:lim h→0+ |h| h = +1

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

 

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

A good way to picture this in your mind is to think:

As I zoom in, does the function tend to become a straight line?

differentiable (zoomed is line) vs not differentiable (zoomed is pointy)

The absolute value function stays pointy even when zoomed in.

Other Reasons

Here are a few more examples:

Floor function  

The Floor and Ceiling Functions are not differentiable at integer values, as there is a discontinuity at each jump. But they are differentiable elsewhere.

 
x^(1/3) slope  

The Cube root function x(1/3)

Its derivative is (1/3)x-(2/3) (by the Power Rule)

At x=0 the derivative is undefined, so x(1/3) is not differentiable.

1/x graph

 

At x=0 the function is not defined so it makes no sense to ask if they are differentiable there.

To be differentiable at a certain point, the function must first of all be defined there!


sin (1/x) graph
 

As we head towards x = 0 the function moves up and down faster and faster, so we cannot find a value it is "heading towards".

So it is not differentiable.

 

Different Domain

But we can change the domain!

absolute positive graph

Example: The function g(x) = |x| with Domain (0,+∞)

The domain is from but not including 0 onwards (all positive values).

Which IS differentiable.

And I am "absolutely positive" about that :)

So the function g(x) = |x| with Domain (0,+∞) is differentiable.

We could also restrict the domain in other ways to avoid x=0 (such as all negative Real Numbers, all non-zero Real Numbers, etc).

 

Why Bother?

Because when a function is differentiable we can use all the power of calculus when working with it.

Continuous

When a function is differentiable it is also continuous.

Differentiable Continuous

But a function can be continuous but not differentiable. For example the absolute value function is actually continuous (though not differentiable) at x=0.