# Product rule for differentiation

ORIGINAL FULL PAGE: Product rule for differentiation
STUDY THE TOPIC AT MULTIPLE LEVELS: Page for school students (first-time learners) | Page for college students (second-time learners) | Page for math majors and others passionate about math |
ALSO CHECK OUT: Practical tips on the topic |Quiz (multiple choice questions to test your understanding) |Pedagogy page (discussion of how this topic is or could be taught)|Page with videos on the topic, both embedded and linked to
View other differentiation rules

## Name

This statement is called the product rule, product rule for differentiation, or Leibniz rule.

## Statement for two functions

### Statement in multiple versions

The product rule is stated in many versions:

Version type Statement
specific point, named functions Suppose  and  are functions of one variable, both of which are differentiable at a real number . Then, the product function , defined as  is also differentiable at , and the derivative at  is given as follows:


or equivalently:


generic point, named functions, point notation Suppose  and  are functions of one variable. Then the following is true wherever the right side expression makes sense (see concept of equality conditional to existence of one side):

generic point, named functions, point-free notation Suppose  and  are functions of one variable. Then, we have the following equality of functions on the domain where the right side expression makes sense (see concept of equality conditional to existence of one side):

We could also write this more briefly as:

Note that the domain of  may be strictly larger than the intersection of the domains of  and , so the equality need not hold in the sense of equality as functions if we care about the domains of definition.
Pure Leibniz notation using dependent and independent variables Suppose  are variables both of which are functionally dependent on . Then:

In terms of differentials Suppose  are both variables functionally dependent on . Then,
.
MORE ON THE WAY THIS DEFINITION OR FACT IS PRESENTED: We first present the version that deals with a specific point (typically with a  subscript) 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

The product rule for differentiation has analogues for one-sided derivatives. More explicitly, we can replace all occurrences of derivatives with left hand derivatives and the statements are true. Alternately, we can replace all occurrences of derivatives with right hand derivatives and the statements are true.

### Partial differentiation

For further information, refer: product rule for partial differentiation

The product rule is also valid if we consider functions of more than one variable and replace the ordinary derivative by the partial derivative, directional derivative, or gradient vector.

## Statement for multiple functions

Below, we formulate the many versions of this product rule:

Version type Statement
specific point, named functions Suppose  are functions defined and differentiable at a point . Then the product  is also differentiable at , and we have:

generic point, named functions, point notation Suppose  are functions. Then the product  satisfies:
 wherever the right side makes sense.
generic points, named functions, point-free notation Suppose  are functions. Then the product  satisfies:
 wherever the right side makes sense. We could also write this more briefly as

Pure Leibniz notation using dependent and independent variables Suppose  are variables functionally dependent on . Then  wherever the right side make sense.
In terms of differentials Suppose  are variables functionally dependent on . Then 

For instance, using the generic point, named functions notation for , we get:



## Reversal for integration

The reverse to this rule, that is helpful for indefinite integrations, is a method called integration by parts.

## Significance

### 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  and  are both differentiable at a point, so is . The one-sided versions allow us to make similar statement for left and right differentiability.
generic point, named functions, point notation This tells us that if both  and  are differentiable on an open interval, then so is . The one-sided versions allow us to make similar statements for closed intervals where we require the appropriate one-sided differentiability at the endpoints.
generic point, point-free notation This can be used to deduce more, namely that the nature of  depends strongly on the nature of  and that of . In particular, if  and  are both continuously differentiable functions on an interval (i.e.,  and  are both continuous on that interval), then  is also continuously differentiable on that interval. This uses the sum theorem for continuity and product theorem for continuity.

### Computational feasibility significance

Each of the versions has its own computational feasibility significance:

Version type Significance
specific point, named functions This tells us that knowledge of the values (in the sense of numerical values)  at a specific point  is sufficient to compute the value of . For instance, if we are given that , we obtain that .
A note on contrast with the (false) freshman product rule: [SHOW MORE]
generic point, named functions This tells us that knowledge of the general expressions for  and  and the derivatives of  and  is sufficient to compute the general expression for the derivative of . See the #Examples section of this page for more examples.

### Computational results significance

Each of the versions has its own computational results significance:

Shorthand Significance What would happen if the freshman product rule were true instead of the product rule?
significance of derivative being zero If  and  are both equal to 0, then so is . In other words, if the tangents to the graphs of  are both horizontal at the point , so is the tangent to the graph of . This result would still hold, but so would a stronger result: namely that if either  or  is zero, so is .
significance of sign of derivative  and  both being positive is not sufficient to ensure that  is positive. However, if all four of  are positive, then  is positive. This is related to the fact that a product of increasing functions need not be increasing. In that case, it would be true that  and  both being positive is sufficient to ensure that  is positive.
significance of uniform bounds  both being uniformly bounded is not sufficient to ensure that  is uniformly bounded. However, if all four functions  are uniformly bounded, then indeed  is uniformly bounded. In that case, it would be true that  and  both uniformly being bounded is sufficient to ensure that  is uniformly bounded.

## Compatibility checks

### Symmetry in the functions being multiplied

We know that the product of two functions is symmetric in them, i.e., . Thus, the product rule for differentiation should satisfy the condition that the formula for  is symmetric in  and , i.e., we get the same formula for . This is indeed true using the commutativity of addition and multiplication:



### Associative symmetry

This is a compatibility check showing that for a product of three functions , we get the same product rule formula whether we associate this product as  or as .

• Associating as :



• Associating as :



### Compatibility with linearity

Consider functions  and the expression:



This can be differentiated in two ways, using the product rule on the left side and then linearity of differentiation, or by differentiating the right side and using the product rule in each term. We verify that both yield the same result:

• Left side: Differentiating, we get 
• Right side: Differentiating, we get .

Thus, both sides are equal and the product rule for differentiation checks out.

### Compatibility with notions of order

This section explains why the product rule is compatible with notions of order  that satisfy:

• 
• 
• If , and both  are positive (in a suitable sense) then  equals it. Even if they are not both positive,  usually has the same order

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.
•  also (plausibly) has order : Note that  has order  and . Adding them should give something of order .

Thus, the product rule is compatible with the order notion.

Note that the freshman product rule is incompatible with notions of order: [SHOW MORE]

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 (though the zero polynomial creates some trouble for multiplication). The notion of positive can be taken as having a positive leading coefficients.
• For functions that are zero at a particular point, the order notion above can be taken as the order of zero at the point.

## Case of infinite or undefined values

For further information, refer: Using the product rule for differentiation for limiting behavior at points with undefined derivative

The product rule for differentiation has analogues for infinities, with the appropriate caveats about indeterminate forms. Specifically, we have the following:

    Conclusion about  Explanation
finite finite undefined undefined insufficient information (could be finite or undefined) We don't know the details behind the undefined
nonzero nonzero and same sign as  vertical tangent vertical tangent of same type as for  (i.e., either both are increasing or both are decreasing) vertical tangent, type (increasing/decreasing) is determined by signs of  and types of vertical tangent for  [SHOW MORE]
nonzero nonzero and opposite sign to  vertical tangent vertical tangent of same type as for  (i.e., either both are increasing or both are decreasing) insufficient information [SHOW MORE]
nonzero nonzero and same sign as  vertical tangent vertical tangent of opposite type as for  (i.e., one is increasing and one is decreasing) insufficient information
nonzero nonzero and opposite sign to  vertical tangent vertical tangent of opposite type as for  (i.e., one is increasing and one is decreasing) vertical tangent, type depends on signs
zero known whether it is zero, positive, or negative known whether it is finite, vertical tangent, etc. vertical tangent insufficient information in all cases.

## Examples

For practical tips and explanations on how to apply the product rule in practice, check out Practical:Product rule for differentiation

### Sanity checks

We first consider examples where the product rule for differentiation confirms something we already knew through other means. In all examples, we assume that both  and  are differentiable functions:

Case The derivative of  Direct justification (without use of product rule) Justification using product rule, i.e., computing it as 
 is the zero function. zero function  for all , so its derivative is also zero . Both  and  are zero functions, so  is everywhere zero.
 is a constant nonzero function with value .  The function is , and the derivative is , because the constant can be pulled out of the differentiation process.  simplifies to . Since  is constant,  is the zero function, hence so is . The sum is thus .
  The derivative is  by the chain rule for differentiation: we are composing the square function and . We get .
 zero function The product is , which is a constant function, so its derivative is zero. We get . By the chain rule, , so plugging in, we get , which simplifies to zero.

### Nontrivial examples where simple alternate methods exist

Here is a simple trigonometric example:

.

### Nontrivial examples where simple alternate methods do not exist

Consider a product of the form:



Using the product rule, we get:



## Proof

There are many different versions of the proof, given below: