WHEN DOES A FUNCTION NOT HAVE A
DERIVATIVE??
SECTION 3.2
A TERM TO KNOW:
•  f(x) is DIFFERENTIABLE at a point if the
derivative exists at that point.
Where does f ‘ (x) NOT exist?
Where does f ‘ (x) NOT exist?
1. a CORNER of a graph.
An example of a function that has a
corner: f (x) = |x| at x=0
An example of a function that has a
corner: f (x) = |x| at x=0
A FUNCTION HAS NO DERIVATIVE AT
A CORNER.
An example of a function that has a
corner: f (x) = |x| at x=0
A FUNCTION HAS NO DERIVATIVE AT
A CORNER. Why???????
Check out the derivative of
f(x)=|x|
f ‘ (x)
Clearly, f ‘ (x) does not exist at x=0 since f ‘ (x) is
not continuous at x=0.
Therefore f (x) =|x| is NOT
differentiable at x=0 since there is a
corner on the graph of f(x) even
though f(x)=|x| is continuous at x=0.
f(x)=|x|
Where does f ‘ (x) NOT exist?
1.  a CORNER of a graph.
Where does f ‘ (x) NOT exist?
1.  a CORNER of a graph.
2.  a CUSP on a graph.
An example of a function with a
cusp is:
f (x) = (x − 1)
2
3
An example of a function with a
cusp is:
f (x) = (x − 1)
2
3
WHAT DOES THE GRAPH OF
f (x) = (x − 1)
2
3
LOOK LIKE????
f (x) = (x − 1)
2
3
f (x) = (x − 1)
2
3
COMPARE f(x) to f ‘(x).
f (x) = (x − 1)
2
3
f ‘ (x)
f (x) = (x − 1)
2
3
f ‘ (x)
Look at the cusp on f (x) at x=1 and what
is happening at x=1 on f ‘(x).
f (x) = (x − 1)
2
3
f ‘ (x)
Since f ‘ (x) is NOT continuous at x=1,
f(x) is not differentiable at x=1 even though
f(x) is continuous at x=1.
30
Where does f ‘ (x) NOT exist?
1.  a CORNER of a graph.
2.  a CUSP on a graph.
Where does f ‘ (x) NOT exist?
1.  a CORNER of a graph of f (x).
2.  a CUSP on a graph of f (x).
3.  a VERTICAL TANGENT on a graph
of f (x).
An example of a function with a vertical
tangent is:
3
f (x) = x
An example of a function with a vertical
tangent is:
3
f (x) = x
f (x) = x
3
Compare f (x) to f ‘(x)
f (x) = 3 x
f ‘ (x)
f (x) = 3 x
f ‘ (x)
Notice how f ‘ (x) is not continuous at x=0?
f (x) = 3 x
f ‘ (x)
This means f (x) is not differentiable at x=0
where f (x) has a vertical tangent even
though f (x) is continuous at x=0.
Where does f ‘ (x) NOT exist?
1.  a CORNER of a graph of f (x).
2.  a CUSP on a graph of f (x).
3.  a VERTICAL TANGENT on a graph
of f (x).
Where does f ‘ (x) NOT exist?
1.  a CORNER of a graph of f (x).
2.  a CUSP on a graph of f (x).
3.  a VERTICAL TANGENT on a graph
of f (x).
4. Any where f(x) itself is not continuous.
This is an important point of differentiability:
A function, f(x) can not have a derivative
in other words, be differentiable, where f(x)
is not continuous, for example……….
1
f (x) =
x−2
at x=2
f(x) is not continuous at x=2, therefore
f(x) is not differentiable at x=2.
BUT…………….
What about piecewise functions????