# Differentiable

Differentiable means that the derivative **exists** ...

### Example: is x^{2} + 6x differentiable?

Derivative rules tell us the derivative of x^{2} is 2x and the derivative of x is 1, so:

Its derivative is **2x + 6**

So yes! x^{2} + 6x is differentiable.

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

## DomainIn its simplest form the domain is |

### Example (continued)

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

For **x ^{2} + 6x**, its derivative of

**2x + 6**exists for all Real Numbers.

So we are still safe: x^{2} + 6x is differentiable.

But what about this:

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

|x| looks like this: |

At **x=0** it has a very pointy change!

**Does the derivative exist at x=0?**

## Testing

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

lim
**h→0**
\frac{f(a+h) − f(a)}{h}

### Example (continued)

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

**h→0**\frac{f(a+h) − f(a)}{h}

**h→0**\frac{|a+h| − |a|}{h}

**h→0**\frac{|h| − |0|}{h}

**h→0**\frac{|h|}{h}

**In fact that limit does not exist!** To see why, let's compare left and right side limits:

**h→0**\frac{|h|}{h} = −1

^{−}**h→0**\frac{|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?

The absolute value function stays pointy at x=0 even when zoomed in.

## Other Reasons

Here are a few more examples:

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

The Cube root function Its derivative is At |

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

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 there. |

## Different Domain

But we can change the domain!

### 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.