Chain rule for differentiation
ORIGINAL FULL PAGE: Chain rule for differentiation
STUDY THE TOPIC AT MULTIPLE LEVELS:
ALSO CHECK OUT: Quiz (multiple choice questions to test your understanding) |Page with videos on the topic, both embedded and linked to
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 for two functions
The chain rule is stated in many versions:
Version type | Statement |
---|---|
specific point, named functions | Suppose ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
generic point, named functions, point notation | Suppose ![]() ![]() ![]() |
generic point, named functions, point-free notation | Suppose ![]() ![]() ![]() ![]() |
pure Leibniz notation | Suppose ![]() ![]() ![]() ![]() ![]() |
MORE ON THE WAY THIS DEFINITION OR FACT IS PRESENTED: We first present the version that deals with a specific point (typically with asubscript) in the domain of the relevant functions, and then discuss the version that deals with a point that is free to move in the domain, by dropping the subscript. Why do we do this?
The purpose of the specific point version is to emphasize that the point is fixed for the duration of the definition, i.e., it does not move around while we are defining the construct or applying the fact. However, the definition or fact applies not just for a single point but for all points satisfying certain criteria, and thus we can get further interesting perspectives on it by varying the point we are considering. This is the purpose of the second, generic point version.
One-sided version
A one-sided version of sorts holds, but we need to be careful, since we want the direction of differentiability of to be the same as the direction of approach of
to
. The following are true:
Condition on ![]() ![]() |
Condition on ![]() ![]() |
Conclusion |
---|---|---|
left differentiable at ![]() |
differentiable at ![]() |
The left hand derivative of ![]() ![]() ![]() ![]() ![]() |
right differentiable at ![]() |
differentiable at ![]() |
The right hand derivative of ![]() ![]() ![]() ![]() ![]() |
left differentiable at ![]() ![]() ![]() |
left differentiable at ![]() |
the left hand derivative is the left hand derivative of ![]() ![]() ![]() ![]() |
right differentiable at ![]() ![]() ![]() |
right differentiable at ![]() |
the right hand derivative is the right hand derivative of ![]() ![]() ![]() ![]() |
left differentiable at ![]() ![]() ![]() |
right differentiable at ![]() |
the left hand derivative is the right hand derivative of ![]() ![]() ![]() ![]() |
right differentiable at ![]() ![]() ![]() |
left differentiable at ![]() |
the right hand derivative is the left hand derivative of ![]() ![]() ![]() ![]() |
Statement for multiple functions
Suppose are functions. Then, the following is true wherever the right side makes sense:
For instance, in the case , we get:
In point notation, this is:
Related rules
Similar facts in single variable calculus
- Chain rule for higher derivatives
- Product rule for differentiation
- Product rule for higher derivatives
- Differentiation is linear
- Inverse function theorem (gives formula for derivative of inverse function).
- Chain rule for differentiation of formal power series
Similar facts in multivariable calculus
Reversal for integration
If a function is differentiated using the chain rule, then retrieving the original function from the derivative typically requires a method of integration called integration by substitution. Specifically, that method of integration targets expressions of the form:
The -substitution idea is to set
and obtain:
We now need to find a function such that
. The integral is
. Plugging back
, we obtain that the indefinite integral is
.
Significance
Why more naive chain rules don't make sense
There are two naive versions of the chain rule one might come up with, neither of which holds:
and
Even without doing any mathematics, we can deduce that neither of these rules can be correct. How? Any rule that holds generically must involve evaluating or
only at points that we know to be in the domain of
. The only such point in this context is
. Therefore, the chain rule cannot involve evaluating
or
at any point other than
.
Note that our actual chain rule:
is quite similar to the naive but false rule , and can be viewed as the corrected version of the rule once we account for the fact that
can only be calculated after transforming
to
.
Qualitative and existential significance
Each of the versions has its own qualitative significance:
Version type | Significance |
---|---|
specific point, named functions | This tells us that if ![]() ![]() ![]() ![]() ![]() ![]() |
generic point, named functions, point notation | If ![]() ![]() ![]() ![]() |
generic point, named functions, point-free notation | We can deduce properties of ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
Computational feasibility significance
Each of the versions has its own computational feasibility significance:
Version type | Significance |
---|---|
specific point, named functions | If we know the values (in the sense of numerical values) ![]() ![]() ![]() |
generic point, named functions | This tells us that knowledge of the general expressions for the derivatives of ![]() ![]() ![]() Note that we do not need to know ![]() ![]() ![]() ![]() ![]() |
Computational results significance
Shorthand | Significance |
---|---|
significance of derivative being zero | If ![]() ![]() ![]() ![]() ![]() ![]() Also, if ![]() ![]() ![]() ![]() Note that it is essential in both cases that the other function be differentiable at the appropriate point. Here are some counterexamples when it's not: [SHOW MORE] |
significance of sign of derivative | The product of the signs of ![]() ![]() ![]() ![]() ![]() |
significance of uniform bounds on derivatives | If ![]() ![]() ![]() ![]() ![]() |
Compatibility checks
Associative symmetry
This is a compatibility check for showing that for a composite of three functions , the formula for the derivative obtained using the chain rule is the same whether we associate it as
or as
.
- Derivative as
. We first apply the chain rule for the pair of functions
and then for the pair of functions
:
In point-free notation:
In point notation (i.e., including a symbol for the point where the function is applied):
- Derivative as
. We first apply the chain rule for the pair of functions
and then for the pair of functions
:
In point-free notation:
In point notation (i.e., including a symbol for the point where the function is applied):
Compatibility with linearity
Consider functions . We have that:
The function can be differentiated either by differentiating the left side or by differentiating the right side. The compatibility check is to ensure that we get the same result from both methods:
- Left side: In point-free notation:
In point notation (i.e., including a symbol for the point of application):
- Right side: In point-free notation:
We get .
In point notation:
Thus, we get the same result on both sides, indicating compatibility.
Note that it is not in general true that , so there is no compatibility check to be made there.
Compatibility with product rule
Consider functions . We have that:
The function can be differentiated either by differentiating the left side or by differentiating the right side. The two processes use the product rule for differentiation in different ways. The compatibility check is to ensure that we get the same result from both methods:
- Left side: In point-free notation:
In point notation:
- Right side: In point-free notation:
In point notation:
Note that it is not in general true that , so no compatibility check needs to be made there.
Compatibility with notions of order
This section explains why the chain rule is compatible with notions of order that satisfy:
Suppose and
. Then we have the following:
-
has order
: First, note that
has order
by the product relation for order. Next, note that differentiating pushes the order down by one.
-
has order
: Note that
has order
and
has order
. Adding, we get <math(m - 1)n + n - 1 = mn - 1</math>.
Some examples of the notion of order which illustrate this are:
- For nonzero polynomials, the order notion above can be taken as the degree of the polynomial.
- For functions that are zero at a particular point, the order notion above can be taken as the order of zero at the point. Note that in this case, the order of zero for
will be calculated at 0 rather than the original point at which
is evaluated.
Examples
Sanity checks
We first consider examples where the chain rule for differentiation confirms something we already knew by other means:
Case on ![]() |
Case on ![]() |
![]() |
Direct justification, without using the chain rule | Justification using the chain rule, i.e., by computing ![]() |
---|---|---|---|---|
a constant function | any differentiable function | zero function | ![]() |
By the chain rule, ![]() ![]() ![]() ![]() ![]() |
any differentiable function | a constant function with value ![]() |
zero function | ![]() ![]() |
By the chain rule, ![]() ![]() ![]() ![]() ![]() |
the identity function, i.e., the function ![]() |
any differentiable function | ![]() |
![]() ![]() |
![]() ![]() ![]() ![]() ![]() ![]() ![]() |
any differentiable function | the identity function | ![]() |
![]() ![]() |
![]() ![]() ![]() ![]() ![]() ![]() |
the square function | any differentiable function | ![]() |
![]() ![]() |
![]() ![]() ![]() ![]() ![]() |
a one-one differentiable function | the inverse function of ![]() |
1 | ![]() ![]() |
![]() ![]() ![]() |
Nontrivial examples
The chain rule is necessary for computing the derivatives of functions whose definition requires one to compose functions. The chain rule still isn't the only option: one can always compute the derivative as a limit of a difference quotient. But it does offer the only option if one restricts oneself to operating within the family of differentiation rules.
Some examples of functions for which the chain rule needs to be used include:
- A trigonometric function applied to a nonlinear algebraic function
- An exponential function applied to a nonlinear algebraic function
- A composite of two trigonometric functions, two exponential functions, or an exponential and a trigonometric function
A few examples are below.
Sine of square function
Consider the sine of square function:
.
We use the chain rule for differentiation viewing the function as the composite of the square function on the inside and the sine function on the outside:
Sine of sine function
Consider the sine of sine function:
The derivative is: