Derivatives of Constant Multiples

Example:

V = 10*(ln(q))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:

(ln(q))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,

(ln(q))²

we can think of as

f(q)

So the derivative is

V '	= (	c	f(q)	)'
	=	c	f '(q)
	=	10	((ln(q))²)'

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

(ln(q))²

2*ln(q) q^-1

(by the chain rule)

so the finished derivative is

V '	=	10	( 2*ln(q) q^-1 )
	=	20*q^-1 ln(q)

additional explanation for the derivative of constant multiples
see another derivative of constant multiples example
practice gateway test
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Page Generated: Wed Jul 29 05:09:06 2026
Comments to Gavin LaRose
glarose@umich.edu
©2001 Gavin LaRose, University of Michigan Math Dept.