Derivatives of Products

Example:
W = (pe + 3) sqrt(6 + p)

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:

W = ( pe + 3) ( sqrt(6 + p))

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,

W = ( pe + 3) ( sqrt(6 + p))
we can think of as
W = f(p) g(p)    
So the derivative is
W ' = ( f(p) g(p) )'    
  = f '(p) g(p) + f(p) g '(p)
  = ( pe + 3 )' ( sqrt(6 + p) ) + ( pe + 3 ) ( sqrt(6 + p) )'
and we just need to know each of the derivatives on the right-hand side of the equation. In this case these are
( pe + 3 )' = e*pe-1 + 0 (by the derivative rule for sums, power rule, and the derivative rule for constants)
( sqrt(6 + p) )' = (1/2)*(6 + p)-1/2 (0 + 1) (by the chain rule)
so the finished derivative is
W ' = ( e*pe-1 + 0 ) ( sqrt(6 + p) ) + ( pe + 3 ) ( (1/2)*(6 + p)-1/2 (0 + 1) )
  = e*pe-1 sqrt(6 + p) + (1/2)*(pe + 3) (6 + p)-1/2
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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.