Derivatives of Sums

Example:
y = sqrt((-2)*x6 + 7*x2) - A*x6

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((-2)*x6 + 7*x2) ) - ( A*x6 )

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((-2)*x6 + 7*x2) ) - ( A*x6 )
we can think of as
y = f(x) - g(x)
So the derivative is
y ' = ( f(x) - g(x) )'
  = f '(x) - g '(x)  
  = ( sqrt((-2)*x6 + 7*x2)) ' - ( A*x6) '  
and we just need to know each of the derivatives on the right-hand side of the equation. In this case these are
( sqrt((-2)*x6 + 7*x2) )' = (1/2)*((-2)*x6 + 7*x2)-1/2 ((-2)*6*x5 + 7*2*x) (by the chain rule)
( A*x6 )' = A*6*x5 (by the rule for constant multiples, and the power rule)
so the finished derivative is
y ' = (1/2)*((-2)*x6 + 7*x2)-1/2 ((-2)*6*x5 + 7*2*x) - A*6*x5  
  = (1/2)*((-2)*x6 + 7*x2)-1/2 ((-12)*x5 + 14*x) - (6A)*x5
[]

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.