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[MUSIC]. 
When I say a function is differentiable, 

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what I really mean is that when I zoom 
in, the function looks like a straight 

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line. 
This is the graph of the absolute value 

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function. 
Let's zoom in on the origin. 

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No matter how much I zoom in, this graph 
doesn't look like a straight line. 

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Consequently, the absolute value function 
is not differentiable at 0. 

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We can also see this from the limit 
definition of derivative. 

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So we just going to name the absolute 
value function f, for the time being. 

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What I am trying to calculate is the 
derivative of f at 0. 

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I want to know is this function 
differentiable at 0. 

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By the definition the derivative is the 
limit. 

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As h approaches 0, of the function, at |0 
+ h| - |0| / h. 

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Now I can simplify that a bit. 
The absolute value of 0 + h, is just the 

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absolute value of h, and the absolute 
value of 0, so I don't even need to 

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subtract 0. 
And I"m dividing by h. 

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Now what's the limit as it approaches 0 
of absolute value h/h. 

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That limit doesn't exist and consequently 
this function's not differential at zero. 

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If you wonder why that limit doesn't 
exist, well think back to our 2 sided 1 

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sided limit discussion from before. 
What's the limit as h approaches 0 from 

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the right-hand side of |h| / h? It's 1. 
Well what's the limit as h approaches 0 

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from the left-hand side of |h| / h? It's 
-1. 

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And 1 is not equal to -1. 
Because the one-sided limits disagree The 

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two sided limit doesn't exist. 
And this limit calculating the derivative 

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means that this function is not 
differentiable at 0. 

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Of course, that raises the question why 
should you care about differentiable 

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functions at all? 
Here's some terrible looking function. 

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But it's differentiable. 
So if I zoom in on some point, the thing 

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looks like a straight line. 
Calculus is all about replacing the 

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curved objects that we can't understand 
with straight lines, which we have some 

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hope of understanding. 
[MUSIC]