Derivatives of Products

Example:
f(t) = (e*t + 7) ln(4*t + 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:

f(t) = ( e*t + 7) ( ln(4*t + 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,

f(t) = ( e*t + 7) ( ln(4*t + 3))
we can think of as
f(t) = g(t) h(t)    
So the derivative is
f '(t) = ( g(t) h(t) )'    
  = g '(t) h(t) + g(t) h '(t)
  = ( e*t + 7 )' ( ln(4*t + 3) ) + ( e*t + 7 ) ( ln(4*t + 3) )'
and we just need to know each of the derivatives on the right-hand side of the equation. In this case these are
( e*t + 7 )' = e + 0 (by the derivative rule for sums, rule for constant multiples, and the derivative rule for constants)
( ln(4*t + 3) )' = (4*t + 3)-1 (4 + 0) (by the chain rule)
so the finished derivative is
f '(t) = ( e + 0 ) ( ln(4*t + 3) ) + ( e*t + 7 ) ( (4*t + 3)-1 (4 + 0) )
  = e*ln(4*t + 3) + 4*(e*t + 7) (4*t + 3)-1
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additional explanation for the product rule
see another product rule example
practice gateway test
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glarose@umich.edu
©2001 Gavin LaRose, University of Michigan Math Dept.