Mathmajor:Product rule for differentiation
For full information, see product rule for differentiation.
ORIGINAL FULL PAGE: Product rule for differentiation
STUDY THE TOPIC AT MULTIPLE LEVELS: Page for school students (firsttime learners)  Page for college students (secondtime 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
Contents
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:

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, pointfree 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.
Onesided version
The product rule for differentiation has analogues for onesided 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, pointfree 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:
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 onesided 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 onesided versions allow us to make similar statements for closed intervals where we require the appropriate onesided differentiability at the endpoints. 
generic point, pointfree 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. 
Proof
There are many different versions of the proof, given below: