Derivatives of Products

Example:
y = (3*x + 8) cos(8*x + 3)

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 = ( 3*x + 8) ( cos(8*x + 3))

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 = ( 3*x + 8) ( cos(8*x + 3))
we can think of as
y = f(x) g(x)    
So the derivative is
y ' = ( f(x) g(x) )'    
  = f '(x) g(x) + f(x) g '(x)
  = ( 3*x + 8 )' ( cos(8*x + 3) ) + ( 3*x + 8 ) ( cos(8*x + 3) )'
and we just need to know each of the derivatives on the right-hand side of the equation. In this case these are
( 3*x + 8 )' = 3 + 0 (by the derivative rule for sums, rule for constant multiples, and the derivative rule for constants)
( cos(8*x + 3) )' = (-1)*sin(8*x + 3) (8 + 0) (by the chain rule)
so the finished derivative is
y ' = ( 3 + 0 ) ( cos(8*x + 3) ) + ( 3*x + 8 ) ( (-1)*sin(8*x + 3) (8 + 0) )
  = 3*cos(8*x + 3) - 8*(3*x + 8) sin(8*x + 3)
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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.