Derivatives of Sums

Example:
B = (-6)*y7 + 2*y5 + 4

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:

B = ( (-6)*y7 ) + ( 2*y5 ) + ( 4 )

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,

B = ( (-6)*y7 ) + ( 2*y5 ) + ( 4 )
we can think of as
B = f(y) + g(y) + h(y)
So the derivative is
B ' = ( f(y) + g(y) + h(y) )'
  = f '(y) + g '(y) + h '(y)  
  = ( (-6)*y7) ' + ( 2*y5) ' + ( 4) '  
and we just need to know each of the derivatives on the right-hand side of the equation. In this case these are
( (-6)*y7 )' = (-6)*7*y6 (by the rule for constant multiples, and the power rule)
( 2*y5 )' = 2*5*y4 (by the rule for constant multiples, and the power rule)
( 4 )' = 0 (by the derivative rule for constants)
so the finished derivative is
B ' = (-6)*7*y6 + 2*5*y4 + 0  
  = (-42)*y6 + 10*y4
[]

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.