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.


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?


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

lim h→0 f(a+h) − f(a) h

Example (continued)

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


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

In fact that 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 at x=0


 f(x) = |x| is not differentiable at x=0

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 at x=0 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, unless we exclude x=0.

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 any value it is "heading towards".

So it is not differentiable there.


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.


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.


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