Derivatives of Sums

Example:
y = ((ln(7))*x)6 + 6*x2 - e

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(7))*x)6 ) + ( 6*x2 ) - ( e )

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(7))*x)6 ) + ( 6*x2 ) - ( e )
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(7))*x)6) ' + ( 6*x2) ' - ( e) '  
and we just need to know each of the derivatives on the right-hand side of the equation. In this case these are
( ((ln(7))*x)6 )' = (ln(7))*6*((ln(7))*x)5 (by the chain rule)
( 6*x2 )' = 6*2*x (by the rule for constant multiples, and the power rule)
( e )' = 0 (by the derivative rule for constants)
so the finished derivative is
y ' = (ln(7))*6*((ln(7))*x)5 + 6*2*x - 0  
  = (6(ln(7)))*((ln(7))*x)5 + 12*x
[]

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.