Derivatives of Sums

Example:
F(y) = (ln(2))*(y4 + 2)7 - 7*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(y) = ( (ln(2))*(y4 + 2)7 ) - ( 7*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(y) = ( (ln(2))*(y4 + 2)7 ) - ( 7*y ) - ( 6 )
we can think of as
F(y) = f(y) - g(y) - h(y)
So the derivative is
F '(y) = ( f(y) - g(y) - h(y) )'
  = f '(y) - g '(y) - h '(y)  
  = ( (ln(2))*(y4 + 2)7) ' - ( 7*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
( (ln(2))*(y4 + 2)7 )' = (ln(2))*7*(y4 + 2)6 (4*y3 + 0) (by the rule for constant multiples, and the chain rule)
( 7*y )' = 7 (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 '(y) = (ln(2))*7*(y4 + 2)6 (4*y3 + 0) - 7 - 0  
  = (28(ln(2)))*y3 (y4 + 2)6 - 7
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additional explanation for the derivative of sums
see another derivative of sums example
practice gateway test
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glarose@umich.edu
©2001 Gavin LaRose, University of Michigan Math Dept.