Derivatives of Sums

Example:
M = (7*y)6 - 7*y + y3 - 4

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:

M = ( (7*y)6 ) - ( 7*y ) + ( y3 ) - ( 4 )

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,

M = ( (7*y)6 ) - ( 7*y ) + ( y3 ) - ( 4 )
we can think of as
M = f(y) - g(y) + h(y) - p(y)
So the derivative is
M ' = ( f(y) - g(y) + h(y) - p(y) )'
  = f '(y) - g '(y) + h '(y) - p '(y)  
  = ( (7*y)6) ' - ( 7*y) ' + ( y3) ' - ( 4) '  
and we just need to know each of the derivatives on the right-hand side of the equation. In this case these are
( (7*y)6 )' = 7*6*(7*y)5 (by the chain rule)
( 7*y )' = 7 (by the rule for constant multiples, and the derivative rule for variables)
( y3 )' = 3*y2 (by the power rule)
( 4 )' = 0 (by the derivative rule for constants)
so the finished derivative is
M ' = 7*6*(7*y)5 - 7 + 3*y2 - 0  
  = 42*(7*y)5 - 7 + 3*y2
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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.