Derivatives of Sums

Example:
y = (ln(2))*(6*x6 - 4)6 - 5*x - 7

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 = ( (ln(2))*(6*x6 - 4)6 ) - ( 5*x ) - ( 7 )

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 = ( (ln(2))*(6*x6 - 4)6 ) - ( 5*x ) - ( 7 )
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)  
  = ( (ln(2))*(6*x6 - 4)6) ' - ( 5*x) ' - ( 7) '  
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))*(6*x6 - 4)6 )' = (ln(2))*6*(6*x6 - 4)5 (6*6*x5 - 0) (by the rule for constant multiples, and the chain rule)
( 5*x )' = 5 (by the rule for constant multiples, and the derivative rule for variables)
( 7 )' = 0 (by the derivative rule for constants)
so the finished derivative is
y ' = (ln(2))*6*(6*x6 - 4)5 (6*6*x5 - 0) - 5 - 0  
  = (216(ln(2)))*x5 (6*x6 - 4)5 - 5
[]

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.