Practical:Product rule for differentiation
This article considers practical aspects of the product rule for differentiation: how is this rule used in actual computations?
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
Contents
Statement to remember
The statement of the product rule for differentiation that we will be using is:
where and .
Procedure to apply the product rule for differentiation
The product rule for differentiation is useful as a technique for differentiating functions that are expressed in the form of products of simpler functions.
Most explicit procedure
The explicit procedure is outlined below:
- Identify the two functions whose product is the given function. In other words, explicitly decompose the function as a product of two functions. We will here call the functions and , though you may choose to give them different names.
- Calculate the derivatives of and separately, on the side.
- Plug into the product rule formula the expressions for the functions and their derivatives.
- Simplify the expression thus obtained (this is optional).
Here is an example of a differentiation problem where we use this explicit procedure:
Differentiate the function with respect to
We proceed step by step:
- Identify the two functions: Define and . Then, by definition.
- Calculate the derivatives: The derivative of is and the derivative of is .
- Plug into the product rule formula: We get .
- Simplify the expression obtained: We get .
Here is another example of a differentiation problem where we use this explicit procedure:
Differentiate the function
- Identify the two functions: Define and
- Calculate the derivatives: The derivative of is and the derivative of is .
- Plug into the product rule formula: We get .
- Simplify the expression obtained: The expression is already simplified. If we wish to collect terms, we can rewrite as .
More inline procedure using Leibniz notation
Although the explicit procedure above is fairly clear, Step (2) of the procedure can be a waste of time in the sense of having to do the derivative calculations separately. If you are more experienced with doing differentiation quickly, you can combine Steps (2) and (3) by calculating the derivatives while plugging into the formula, rather than doing the calculations separately prior to plugging into the formula. Further, we do not need to explicitly name the functions if we use the Leibniz notation to compute the derivatives inline.
The shorter procedure is outlined below:
- Identify the two functions being multiplied (but you don't have to give them names).
- Plug into the formula for the product rule, using the Leibniz notation for derivatives that have not yet been computed.
- Compute derivatives and simplify
For instance, consider the problem:
Differentiate the function
The procedure is:
- Identify the two functions: The functions are and .
- Plug into the formula for the product rule: We get:
- Compute derivatives and simplify: we get:
. The expression can be expanded and simplified if desired.
Shortest inline procedure
If you are really experienced with doing derivatives in your head, you can shorten the procedure even further by combining Steps (2) and (3) in the previous procedure. The procedure has two steps:
- Identify the two functions being multiplied (but you don't have to give them names).
- Use the formula for the product rule, computing the derivatives of the functions while plugging them into the formula
For instance, going back to the example used in the beginning:
Differentiate the function with respect to
This can be done quickly:
- Identify the two functions: They are and .
- Use the formula for the product rule, computing the derivatives of the functions while plugging them into the formula: We get .
Choosing between procedures
The procedures are not fundamentally different, but they differ in the degree of explicitness of the steps. Generally speaking, the following are recommended:
- If the functions being multiplied are fairly easy to differentiate mentally, use the shortest inline procedure -- this is fast and reliable.
- If the functions being multiplied are somewhat more difficult to differentiate, then choose between the other two more explicit procedures, based on whether you are more comfortable with writing large inline expressions or with doing separate work on the side.
Error types
Incorrect formula
A common mistake in differentiating products of functions is the freshman product rule, i.e., the false rule that the derivative of the product is the product of the derivatives. The good news is that, generally speaking, it is easy to avoid this rule once you have enough experience with the actual product rule.
Writing only one piece of the product rule
This is an error of the incomplete task form and is harder to avoid. What happens here is that you forget to write one of the two pieces being added for the product rule, so perhaps you end up doing:
or
A slight variant involves forgetting one of the factors being multiplied:
Why this error occurs: Usually, this error is common if you are trying to use the shortest inline procedure, i.e., differentiating the functions and applying the product rule simultaneously, and one of the functions being differentiated is rather tricky to differentiate, requiring a product rule or chain rule for differentiation in and of itself.
How to avoid this error:
- When the functions being differentiated are tricky to differentiate, use either the fully explicit procedure or the inline procedure with Leibniz notation. Do not try to simultaneously differentiate the pieces and use the product rule.
- After finishing a product rule problem, ask the following sanity check question: did I get a sum of two products after the application of the product rule? If the answer is no, then check your work.
How to remember the formula
Different versions of the formula
Recall that we stated the product rule as:
The right side could be written in any of eight equivalent ways:
All of these are equivalent, so in some sense it does not matter which of the versions you choose to remember. However, the first version is somewhat preferable to the others for a number of reasons mentioned below:
- The order of multiplication is the same as in the expression being differentiated: the -part is on the left and the -part is on the right. This is true only of versions (1) and (5). This is particularly important when we consider generalization to non-commutative situations such as product rule for differentiation of cross product.
- The differentiation symbol (the prime) hops from left to right: Remembering the rule this way makes it easier to generalize to differentiating products of more than two functions (see product rule for differentiation#Statement for multiple functions).
Quick rationalizations for the formula
If you are a little shaky about the product rule for differentiation, how do you do a reality check on the formula? A quick, intuitive version of the proof of product rule for differentiation using chain rule for partial differentiation will help. Basically, what it says is that to determine how the product changes, we need to count the contributions of each factor being multiplied, keeping the other constant. The contribution of keeping constant is and the contribution of keeping constant is .
Applying the product rule for differentiation to compute values at a point
Knowledge of the values (in the sense of numerical values) at a specific point is sufficient to compute the value of .
For instance, suppose we are given that , we obtain that .