[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]