Derivatives of Sums

Example:
y = t3 - 3*t2 + 4*t - 2 - 2(ln(2))

The first thing to notice when finding the derivative of this function is that it is the sum of several terms, as shown in color below:

y = ( t3 ) - ( 3*t2 ) + ( 4*t ) - ( 2 ) - ( 2(ln(2)) )

The Derivative Rule for Sums:

The derivative of a sum is the sum of the derivatives.
If
  z = ( f(x) + g(x) )
then the derivative of z is
  z' = ( f(x) + g(x) )'
    = f '(x) + g'(x)

So our example,

y = ( t3 ) - ( 3*t2 ) + ( 4*t ) - ( 2 ) - ( 2(ln(2)) )
we can think of as
y = f(t) - g(t) + h(t) - p(t) - q(t)
So the derivative is
y ' = ( f(t) - g(t) + h(t) - p(t) - q(t) )'
  = f '(t) - g '(t) + h '(t) - p '(t) - q '(t)  
  = ( t3) ' - ( 3*t2) ' + ( 4*t) ' - ( 2) ' - ( 2(ln(2))) '  
and we just need to know each of the derivatives on the right-hand side of the equation. In this case these are
( t3 )' = 3*t2 (by the power rule)
( 3*t2 )' = 3*2*t (by the rule for constant multiples, and the power rule)
( 4*t )' = 4 (by the rule for constant multiples, and the derivative rule for variables)
( 2 )' = 0 (by the derivative rule for constants)
( 2(ln(2)) )' = 0 (by the derivative rule for constants)
so the finished derivative is
y ' = 3*t2 - 3*2*t + 4 - 0 - 0  
  = 3*t2 - 6*t + 4
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additional explanation for the derivative of sums
see another derivative of sums example
practice gateway test
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Comments to Gavin LaRose
glarose@umich.edu
©2001 Gavin LaRose, University of Michigan Math Dept.