Derivatives of Sums

Example:
R(q) = ((-4)*q)7 + (ln(2))*q2

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:

R(q) = ( ((-4)*q)7 ) + ( (ln(2))*q2 )

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,

R(q) = ( ((-4)*q)7 ) + ( (ln(2))*q2 )
we can think of as
R(q) = f(q) + g(q)
So the derivative is
R '(q) = ( f(q) + g(q) )'
  = f '(q) + g '(q)  
  = ( ((-4)*q)7) ' + ( (ln(2))*q2) '  
and we just need to know each of the derivatives on the right-hand side of the equation. In this case these are
( ((-4)*q)7 )' = (-4)*7*((-4)*q)6 (by the chain rule)
( (ln(2))*q2 )' = (ln(2))*2*q (by the rule for constant multiples, and the power rule)
so the finished derivative is
R '(q) = (-4)*7*((-4)*q)6 + (ln(2))*2*q  
  = (-28)*((-4)*q)6 + (2(ln(2)))*q
[]

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.