Derivatives of Sums

Example:
y = (2*x - 1)8 - x + e

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 = ( (2*x - 1)8 ) - ( x ) + ( e )

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 = ( (2*x - 1)8 ) - ( x ) + ( e )
we can think of as
y = f(x) - g(x) + h(x)
So the derivative is
y ' = ( f(x) - g(x) + h(x) )'
  = f '(x) - g '(x) + h '(x)  
  = ( (2*x - 1)8) ' - ( x) ' + ( e) '  
and we just need to know each of the derivatives on the right-hand side of the equation. In this case these are
( (2*x - 1)8 )' = 8*(2*x - 1)7 (2 - 0) (by the chain rule)
( x )' = 1 (by the derivative rule for variables)
( e )' = 0 (by the derivative rule for constants)
so the finished derivative is
y ' = 8*(2*x - 1)7 (2 - 0) - 1 + 0  
  = 16*(2*x - 1)7 - 1
[]

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.