Derivatives of Sums

Example:
y = ((-7)*x)7 - 3*x2 - 4*x - 1

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 = ( ((-7)*x)7 ) - ( 3*x2 ) - ( 4*x ) - ( 1 )

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 = ( ((-7)*x)7 ) - ( 3*x2 ) - ( 4*x ) - ( 1 )
we can think of as
y = f(x) - g(x) - h(x) - p(x)
So the derivative is
y ' = ( f(x) - g(x) - h(x) - p(x) )'
  = f '(x) - g '(x) - h '(x) - p '(x)  
  = ( ((-7)*x)7) ' - ( 3*x2) ' - ( 4*x) ' - ( 1) '  
and we just need to know each of the derivatives on the right-hand side of the equation. In this case these are
( ((-7)*x)7 )' = (-7)*7*((-7)*x)6 (by the chain rule)
( 3*x2 )' = 3*2*x (by the rule for constant multiples, and the power rule)
( 4*x )' = 4 (by the rule for constant multiples, and the derivative rule for variables)
( 1 )' = 0 (by the derivative rule for constants)
so the finished derivative is
y ' = (-7)*7*((-7)*x)6 - 3*2*x - 4 - 0  
  = (-49)*((-7)*x)6 - 6*x - 4
[]

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.