Derivatives of Sums

Example:
z = 5*y7 - 4*y4 + 1 + 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:

z = ( 5*y7 ) - ( 4*y4 ) + ( 1 + 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,

z = ( 5*y7 ) - ( 4*y4 ) + ( 1 + ln(2) )
we can think of as
z = f(y) - g(y) + h(y)
So the derivative is
z ' = ( f(y) - g(y) + h(y) )'
  = f '(y) - g '(y) + h '(y)  
  = ( 5*y7) ' - ( 4*y4) ' + ( 1 + 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
( 5*y7 )' = 5*7*y6 (by the rule for constant multiples, and the power rule)
( 4*y4 )' = 4*4*y3 (by the rule for constant multiples, and the power rule)
( 1 + ln(2) )' = 0 + 0 (by the derivative rule for sums, derivative rule for constants, and the derivative rule for constants (again))
so the finished derivative is
z ' = 5*7*y6 - 4*4*y3 + 0 + 0  
  = 35*y6 - 16*y3
[]

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.