Derivatives of Products

Example:
g(s) = (A*s + 1) ln(s + 1)

The first thing to notice when finding the derivative of this function is that it is the product of several terms, as shown in color below:

g(s) = ( A*s + 1) ( ln(s + 1))

The Derivative Rule for Products:

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 our example,

g(s) = ( A*s + 1) ( ln(s + 1))
we can think of as
g(s) = f(s) h(s)    
So the derivative is
g '(s) = ( f(s) h(s) )'    
  = f '(s) h(s) + f(s) h '(s)
  = ( A*s + 1 )' ( ln(s + 1) ) + ( A*s + 1 ) ( ln(s + 1) )'
and we just need to know each of the derivatives on the right-hand side of the equation. In this case these are
( A*s + 1 )' = A + 0 (by the derivative rule for sums, rule for constant multiples, and the derivative rule for constants)
( ln(s + 1) )' = (s + 1)-1 (1 + 0) (by the chain rule)
so the finished derivative is
g '(s) = ( A + 0 ) ( ln(s + 1) ) + ( A*s + 1 ) ( (s + 1)-1 (1 + 0) )
  = A*ln(s + 1) + (A*s + 1) (s + 1)-1
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additional explanation for the product rule
see another product rule example
practice gateway test
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Comments to Gavin LaRose
glarose@umich.edu
©2001 Gavin LaRose, University of Michigan Math Dept.