Derivatives of Sums

Example:

z = (sqrt(2))*y + sqrt(2) - (ln(2))*y²

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:

( (sqrt(2))*y )

( sqrt(2) )

( (ln(2))*y² )

The Derivative Rule for Sums:

The derivative of a sum is the sum of the derivatives.
If

( f(x)

g(x) )

then the derivative of z is

	z'	=	( f(x)	+	g(x) )'
		=	f '(x)	+	g'(x)

So our example,

( (sqrt(2))*y )

( sqrt(2) )

( (ln(2))*y² )

we can think of as

f(y)

g(y)

h(y)

So the derivative is

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))*y )'	=	(sqrt(2))	(by the rule for constant multiples, and the derivative rule for variables)
( sqrt(2) )'	=	0	(by the derivative rule for constants)
( (ln(2))*y² )'	=	(ln(2))2y	(by the rule for constant multiples, and the power rule)

so the finished derivative is

z '	=	(sqrt(2))	+	0	-	(ln(2))2y
	=	(sqrt(2)) - (2(ln(2)))*y

Page Generated: Tue Apr 7 13:41:15 2026
Comments to Gavin LaRose
glarose@umich.edu
©2001 Gavin LaRose, University of Michigan Math Dept.