[MUSIC]. 
When I say a function is differentiable, 
what I really mean is that when I zoom 
in, the function looks like a straight 
line. 
This is the graph of the absolute value 
function. 
Let's zoom in on the origin. 
No matter how much I zoom in, this graph 
doesn't look like a straight line. 
Consequently, the absolute value function 
is not differentiable at 0. 
We can also see this from the limit 
definition of derivative. 
So we just going to name the absolute 
value function f, for the time being. 
What I am trying to calculate is the 
derivative of f at 0. 
I want to know is this function 
differentiable at 0. 
By the definition the derivative is the 
limit. 
As h approaches 0, of the function, at |0 
+ h| - |0| / h. 
Now I can simplify that a bit. 
The absolute value of 0 + h, is just the 
absolute value of h, and the absolute 
value of 0, so I don't even need to 
subtract 0. 
And I"m dividing by h. 
Now what's the limit as it approaches 0 
of absolute value h/h. 
That limit doesn't exist and consequently 
this function's not differential at zero. 
If you wonder why that limit doesn't 
exist, well think back to our 2 sided 1 
sided limit discussion from before. 
What's the limit as h approaches 0 from 
the right-hand side of |h| / h? It's 1. 
Well what's the limit as h approaches 0 
from the left-hand side of |h| / h? It's 
-1. 
And 1 is not equal to -1. 
Because the one-sided limits disagree The 
two sided limit doesn't exist. 
And this limit calculating the derivative 
means that this function is not 
differentiable at 0. 
Of course, that raises the question why 
should you care about differentiable 
functions at all? 
Here's some terrible looking function. 
But it's differentiable. 
So if I zoom in on some point, the thing 
looks like a straight line. 
Calculus is all about replacing the 
curved objects that we can't understand 
with straight lines, which we have some 
hope of understanding. 
[MUSIC]