Differentiation rule for piecewise definition by interval
This article is about a differentiation rule, i.e., a rule for differentiating a function expressed in terms of other functions whose derivatives are known.
View other differentiation rules
Contents
Statement
Everywhere version
Suppose and
are functions of one variable, such that both of the functions are defined and differentiable everywhere. Consider a function
, defined as follows:
Then, we have the following for continuity:
- The left hand limit of
at
equals
.
- The right hand limit of
at
equals
.
-
is left continuous at
iff
.
-
is right continuous at
iff
.
-
is continuous at
iff
.
We have the following for differentiability:
-
is left differentiable at
iff
, and in this case, the left hand derivative equals
.
-
is right differentiable at
iff
, and in this case, the right hand derivative equals
.
-
is differentiable at
iff (
and
), and in this case, the derivative equals the equal values
and
.
Piecewise definition of derivative
If the conditions for differentiability at are violated, we get the following piecewise definition for
, which excludes the point
from its domain:
If the conditions for differentiability at are satisfied, we get the following piecewise definition for
, which includes the point
in its domain:
where . In particular, the value at
can be included in either the left side or the right side definition.
Version for higher derivatives
Suppose and
are functions of one variable, such that both of the functions are defined and
times differentiable everywhere (and hence in particular the functions and their first
derivatives are continuous), for some positive integer
. Consider the function:
Then, is
times differentiable at
if we have all these conditions:
,
,
,
. In other words, the values should match, and the values of each of the derivatives up to the
derivative should match. In that case, the
derivative of
at
equals the equal values
.
The general piecewise definition of is, in this case:
where .
Local generalization
The above holds with the following modification: we only require to be defined as
on the immediate left of
(i.e., on some interval of the form
for
and as
on the immediate right of
(i.e., on some interval of the form
for
). Further, we only require that
and
be defined and differentiable on open intervals containing
, not necessarily on all of
.
Examples
Example of piecewise rational function
This example is covered in the video embedded above.
Consider the function:
Note that here, in the notation we have used, we have:
Note that the function is defined around zero, i.e., the definition extends to the point zero and the immediate right -- in fact,
is defined and infinitely differentiable on the interval
.
Similarly, is defined around zero, i.e., i.e., the definition extends to the immediate left of zero -- in fact,
is defined and infinitely differentiable on the interval
.
Thus, we see that:
-
,
, and
. Thus, we see that
, so the function
is continuous at 0.
- We have
and
. We see that
and
. We see that
, so
is not differentiable at 0.
This means that the first and higher derivatives of do not exist at 0.
Example of piecewise polynomial function
This example is covered in the video embedded above.
Consider the function:
Here:
We see that:
-
,
, and
. Thus,
, so
is continuous at 0.
-
and
. Evaluated at 0, we get
and
, so
. So,
is not differentiable at 0.
Example of piecewise polynomial function: higher derivatives
This example is covered in the video embedded above.
Consider the function:
Here, . To keep track of what we're doing, we make a table:
Expression for ![]() |
Value for ![]() |
Expression for ![]() |
Value for ![]() |
Conclusion | Explanation | |
---|---|---|---|---|---|---|
Function | ![]() |
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We are also given that the function value at 0 is 0. Thus, ![]() ![]() |
First derivative | ![]() |
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We already checked continuity, and we have now checked that ![]() |
Second derivative | ![]() |
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We already checked differentiability. Thus, it suffices to check that ![]() |
Third derivative | ![]() |
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We have ![]() |
Note that no higher derivative of exists at zero. For instance, we do have that
, but
does not exist.
Here are the explicit piecewise definitions for the derivatives of :
Note that is not defined at 0.
For , we have:
But does not exist.
Caveat
In situations where the definitions given on one side of a point do not extend naturally to the point, we cannot use the above methods. In most such cases, we need to go back to the original definition of the derivative as a limit of a difference quotient.