Derivatives of Sums

Example:
y = sqrt(x4 - x) + 5*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 = ( sqrt(x4 - x) ) + ( 5*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 = ( sqrt(x4 - x) ) + ( 5*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)  
  = ( sqrt(x4 - x)) ' + ( 5*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
( sqrt(x4 - x) )' = (1/2)*(x4 - x)-1/2 (4*x3 - 1) (by the chain rule)
( 5*x )' = 5 (by the rule for constant multiples, and the derivative rule for variables)
( e )' = 0 (by the derivative rule for constants)
so the finished derivative is
y ' = (1/2)*(x4 - x)-1/2 (4*x3 - 1) + 5 + 0  
  = (1/2)*(x4 - x)-1/2 (4*x3 - 1) + 5
[]

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.