Derivatives of Sums

Example:
y = (10*x + 8)e - x-5

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 = ( (10*x + 8)e ) - ( x-5 )

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 = ( (10*x + 8)e ) - ( x-5 )
we can think of as
y = f(x) - g(x)
So the derivative is
y ' = ( f(x) - g(x) )'
  = f '(x) - g '(x)  
  = ( (10*x + 8)e) ' - ( x-5) '  
and we just need to know each of the derivatives on the right-hand side of the equation. In this case these are
( (10*x + 8)e )' = e*(10*x + 8)e-1 (10 + 0) (by the chain rule)
( x-5 )' = (-5)*x-6 (by the power rule)
so the finished derivative is
y ' = e*(10*x + 8)e-1 (10 + 0) - (-5)*x-6  
  = (10e)*(10*x + 8)e-1 + 5*x-6
[]


additional explanation for the derivative of sums
see another derivative of sums example
practice gateway test
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Comments to Gavin LaRose
glarose@umich.edu
©2001 Gavin LaRose, University of Michigan Math Dept.