Derivatives of Products

Example:
y = s2 tan(s)

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:

y = ( s2) ( tan(s))

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,

y = ( s2) ( tan(s))
we can think of as
y = f(s) g(s)    
So the derivative is
y ' = ( f(s) g(s) )'    
  = f '(s) g(s) + f(s) g '(s)
  = ( s2 )' ( tan(s) ) + ( s2 ) ( tan(s) )'
and we just need to know each of the derivatives on the right-hand side of the equation. In this case these are
( s2 )' = 2*s (by the power rule)
( tan(s) )' = 1 / (cos(s))2 (by the derivative rules for basic functions)
so the finished derivative is
y ' = ( 2*s ) ( tan(s) ) + ( s2 ) ( 1 / (cos(s))2 )
  = 2*s tan(s) + s2 (1 / (cos(s))2)
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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.