Derivatives of Sums

Example:
f(s) = sqrt(3*s5 - s2) + 2*s4 - ln(2)

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:

f(s) = ( sqrt(3*s5 - s2) ) + ( 2*s4 ) - ( ln(2) )

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,

f(s) = ( sqrt(3*s5 - s2) ) + ( 2*s4 ) - ( ln(2) )
we can think of as
f(s) = g(s) + h(s) - p(s)
So the derivative is
f '(s) = ( g(s) + h(s) - p(s) )'
  = g '(s) + h '(s) - p '(s)  
  = ( sqrt(3*s5 - s2)) ' + ( 2*s4) ' - ( ln(2)) '  
and we just need to know each of the derivatives on the right-hand side of the equation. In this case these are
( sqrt(3*s5 - s2) )' = (1/2)*(3*s5 - s2)-1/2 (3*5*s4 - 2*s) (by the chain rule)
( 2*s4 )' = 2*4*s3 (by the rule for constant multiples, and the power rule)
( ln(2) )' = 0 (by the derivative rule for constants)
so the finished derivative is
f '(s) = (1/2)*(3*s5 - s2)-1/2 (3*5*s4 - 2*s) + 2*4*s3 - 0  
  = (1/2)*(3*s5 - s2)-1/2 (15*s4 - 2*s) + 8*s3
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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.