Derivatives of Sums

Example:
P = sqrt((-2)*q + 5) - 3*q5 - (p)3

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:

P = ( sqrt((-2)*q + 5) ) - ( 3*q5 ) - ( (p)3 )

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,

P = ( sqrt((-2)*q + 5) ) - ( 3*q5 ) - ( (p)3 )
we can think of as
P = f(q) - g(q) - h(q)
So the derivative is
P ' = ( f(q) - g(q) - h(q) )'
  = f '(q) - g '(q) - h '(q)  
  = ( sqrt((-2)*q + 5)) ' - ( 3*q5) ' - ( (p)3) '  
and we just need to know each of the derivatives on the right-hand side of the equation. In this case these are
( sqrt((-2)*q + 5) )' = (1/2)*((-2)*q + 5)-1/2 (-2 + 0) (by the chain rule)
( 3*q5 )' = 3*5*q4 (by the rule for constant multiples, and the power rule)
( (p)3 )' = 0 (by the derivative rule for constants)
so the finished derivative is
P ' = (1/2)*((-2)*q + 5)-1/2 (-2 + 0) - 3*5*q4 - 0  
  = (-1)*((-2)*q + 5)-1/2 - 15*q4
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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.