Derivatives of Products

As usual, to take the derivative of a product we need to

  1. recognize the product so that we apply the correct rule, and
  2. apply the rule by finding the derivatives we need and plugging them into the rule.
There are really only two parts of the product rule that are tricky: remembering to use it when we see a product, and remembering that the derivative ends up as a sum of several terms. Let's step through a couple of examples to see how this works.

a few popular functions
  power : x r
  polynomial : c n x n + c n-1 x n-1 + ... + c 0
  exponential : a x
  natural log : ln(x)
  sine : sin(x)
  cosine : cos(x)
  tangent : tan(x)
Recognize the Product: First, we look for products of functions. Some popular functions that may show up in such products are shown in the box to the right. Examples of such products are
sin(x) cos(x),   x ex,   x (x2 + 1) tan(x),   and maybe even 5 ln(x).
The first three of these are obviously products -- there are a functions multiplied together, as shown in color here:
sin(x) cos(x) ,  x ex ,  x (x2 + 1) tan(x) ,  and   5 ln(x) .
In the last one the first function is the constant 5. This is a function, so we can use the product rule, but it would be easier to find the derivative with the rule for constant multiples.

Let's write out the functions in these products explicitly in a table. Notice that (1) we're labeling the functions in the products f(x) and g(x) to make them look like the product rule (as given below), and (2) the third of them is a product of 3 functions, so we have an h(x) as well.

f(x) g(x)     f(x)     g(x)  
sin(x) cos(x)     sin(x)     cos(x)  
x ex     x     ex  
5 ln(x)     5     ln(x)  
f(x) g(x) h(x)     f(x)     g(x)     h(x)
x (x2 + 1) tan(x)     x     x2 + 1     tan(x)

Applying the rule: Once we've seen that we're working with a product, we apply the product rule:

The derivative of a product is the derivative of the first term times the second (and third, etc.) term(s), plus the first (and third, etc.) term(s) times the derivative of the second, etc.
If
  z = (f(x) g(x))
then the derivative of z is
  z ' = (f(x) g(x))'
    = f '(x) g(x) + f(x) g '(x)

So to work out the derivative of a product, we need to

  1. find the derivative of each term
  2. take each times the undifferentiated other terms, and then add each of these together.
Be careful with the second of these steps! The first step says find the derivatives of the terms in the product. Let's add these as new columns in our table:

f(x) g(x)     f(x)     g(x)           f '(x)     g '(x)  
sin(x) cos(x)     sin(x)     cos(x)           cos(x)     -sin(x)  
x ex     x     ex           1     ex  
5 ln(x)     5     ln(x)           0     1/x  
f(x) g(x) h(x)     f(x)     g(x)     h(x)     f '(x)     g '(x)     h '(x)
x (x2 + 1) tan(x)     x     x2 + 1     tan(x)     1     2x     1/cos2(x)
Now that we have the derivatives of the terms in the products, we go to step two and assemble the derivative using the product rule:
( f(x) g(x) ) ' = f '(x) g(x) + f(x) g '(x)
( sin(x) cos(x) ) ' = cos(x) cos(x) + sin(x) (-sin(x))
( x ex ) ' = 1 ex + x ex
( 5 ln(x) ) ' = 0 ln(x) + 5 (1/ x)
and
( f(x) g(x) h(x) ) ' = f '(x) g(x) h(x) + f(x) g '(x) h(x) + f(x) g(x) h '(x)
( x (x2+1) tan(x) ) ' = 1 (x2+1) tan(x) + x 2 x tan(x) + x (x2+1) (1/ cos2(x)

Summary: Let's summarize the steps we take to find derivatives of products.

  1. Recognize the product: first we find the functions that were being multiplied together.
  2. Find the derivatives we need: then we find the derivatives of each of these functions.
  3. Assemble the derivative: and then we use the product rule to assemble the derivative of the product.

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see a product rule example
practice test on the product rule
practice gateway test
previous page
Deriv Tutorials: Products
Last Modified: Mon May 7 12:57:55 EDT 2001
Comments to glarose@umich.edu
©2001 Gavin LaRose, UM Math Dept.