Derivatives of Constant Multiples

Example:

B(x) = 5*(sin(x))²

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:

B(x)

(sin(x))²

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,

B(x)

(sin(x))²

we can think of as

B(x)

f(x)

So the derivative is

B '(x)	= (	c	f(x)	)'
	=	c	f '(x)
	=	5	((sin(x))²)'

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

(sin(x))²

2*sin(x) cos(x)

(by the chain rule)

so the finished derivative is

B '(x)	=	5	( 2*sin(x) cos(x) )
	=	10*sin(x) cos(x)

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

Page Generated: Thu Jan 22 05:51:08 2026
Comments to Gavin LaRose
glarose@umich.edu
©2001 Gavin LaRose, University of Michigan Math Dept.