Product rule for differentiation
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
- 1 Name
- 2 Statement for two functions
- 3 Statement for multiple functions
- 4 Related rules
- 5 Reversal for integration
- 6 Significance
- 7 Case of infinite or undefined values
- 8 Examples
- 9 Proof
This statement is called the product rule, product rule for differentiation, or Leibniz rule.
Statement for two functions
If two (possibly equal) functions are differentiable at a given real number, then their pointwise product is also differentiable at that number and the derivative of the product is the sum of two terms: the derivative of the first function times the second function and the first function times the derivative of the second function.
Statement with symbols
The product rule is stated in many versions:
|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:|
|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:|
|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: |
|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.
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.
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.
Statement for multiple functions
Below, we formulate the many versions of this product rule:
|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.
|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:
Similar rules in single variable calculus
- Differentiation is linear: The derivative of the sum is the sum of the derivatives, and scalars can be pulled out of differentiation.
- Chain rule for differentiation
- Product rule for higher derivatives
- Chain rule for higher derivatives
Similar rules in multivariable calculus
- Product rule for differentiation of dot product
- Product rule for differentiation of cross product
- Product rule for differentiation of scalar triple product
Reversal for integration
The reverse to this rule, that is helpful for indefinite integrations, is a method called integration by parts.
Qualitative and existential significance
Each of the versions has its own qualitative 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:
|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|
|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.|
Order of zero and order of growth
This section explains why the product rule is compatible with notions of order that satisfy:
- If , then .
- 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 .
Case of infinite or undefined values
The product rule for differentiation has analogues for infinities, with the appropriate caveats about indeterminate forms. Specifically, we have the following:
|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.|
We first consider examples where the product rule for differentiation confirms something we already knew through other means:
|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 .|
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:
There are many different versions of the proof, given below: