Derivatives of Products

Example:
y = ((p)*x + 5)2 (x2 + 3*x)

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 = ( ((p)*x + 5)2) ( x2 + 3*x)

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 = ( ((p)*x + 5)2) ( x2 + 3*x)
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)
  = ( ((p)*x + 5)2 )' ( x2 + 3*x ) + ( ((p)*x + 5)2 ) ( x2 + 3*x )'
and we just need to know each of the derivatives on the right-hand side of the equation. In this case these are
( ((p)*x + 5)2 )' = 2*((p)*x + 5) (p + 0) (by the chain rule)
( x2 + 3*x )' = 2*x + 3 (by the derivative rule for sums, power rule, and the rule for constant multiples)
so the finished derivative is
y ' = ( 2*((p)*x + 5) (p + 0) ) ( x2 + 3*x ) + ( ((p)*x + 5)2 ) ( 2*x + 3 )
  = (2p)*((p)*x + 5) (x2 + 3*x) + ((p)*x + 5)2 (2*x + 3)
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additional explanation for the product rule
see another product rule example
practice gateway test
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©2001 Gavin LaRose, University of Michigan Math Dept.