Derivatives of Sums

Example:
F = (-4)*(y5 + 1)7 - (sqrt(5))*y + 6

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:

F = ( (-4)*(y5 + 1)7 ) - ( (sqrt(5))*y ) + ( 6 )

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,

F = ( (-4)*(y5 + 1)7 ) - ( (sqrt(5))*y ) + ( 6 )
we can think of as
F = f(y) - g(y) + h(y)
So the derivative is
F ' = ( f(y) - g(y) + h(y) )'
  = f '(y) - g '(y) + h '(y)  
  = ( (-4)*(y5 + 1)7) ' - ( (sqrt(5))*y) ' + ( 6) '  
and we just need to know each of the derivatives on the right-hand side of the equation. In this case these are
( (-4)*(y5 + 1)7 )' = (-4)*7*(y5 + 1)6 (5*y4 + 0) (by the rule for constant multiples, and the chain rule)
( (sqrt(5))*y )' = (sqrt(5)) (by the rule for constant multiples, and the derivative rule for variables)
( 6 )' = 0 (by the derivative rule for constants)
so the finished derivative is
F ' = (-4)*7*(y5 + 1)6 (5*y4 + 0) - (sqrt(5)) + 0  
  = (-140)*y4 (y5 + 1)6 - (sqrt(5))
[]

additional explanation for the derivative of sums
see another derivative of sums example
practice gateway test
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©2001 Gavin LaRose, University of Michigan Math Dept.