Derivatives of Constant Multiples

Example:

h(t) = 4*sin(sqrt(t))

The first thing to notice when finding the derivative of this function is that it is the product of a constant and another function, as shown in color below:

h(t)

sin(sqrt(t))

The Derivative Rule for Constant Multiples:

The derivative of a constant multiple is the constant times thederivative of the function.
If

( f(x) )

then the derivative of z is

	z'	=	( c	f(x) )'
		=	c	f '(x)

So our example,

h(t)

sin(sqrt(t))

we can think of as

h(t)

f(t)

So the derivative is

h '(t)	= (	c	f(t)	)'
	=	c	f '(t)
	=	4	(sin(sqrt(t)))'

and we just need to know the derivative on the right-hand side of the equation. In this case this is

sin(sqrt(t))

(1/2)*cos(sqrt(t)) t^-1/2

(by the chain rule)

so the finished derivative is

h '(t)	=	4	( (1/2)*cos(sqrt(t)) t^-1/2 )
	=	2*t^-1/2 cos(sqrt(t))

additional explanation for the derivative of constant multiples
see another derivative of constant multiples example
practice gateway test
previous page

Page Generated: Mon Oct 20 00:19:56 2025
Comments to Gavin LaRose
glarose@umich.edu
©2001 Gavin LaRose, University of Michigan Math Dept.