Derivatives of Constant Multiples

Example:

y = 8*sqrt(ln(x))

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:

sqrt(ln(x))

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,

sqrt(ln(x))

we can think of as

f(x)

So the derivative is

y '	= (	c	f(x)	)'
	=	c	f '(x)
	=	8	(sqrt(ln(x)))'

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

sqrt(ln(x))

(1/2)*(ln(x))^-1/2 x^-1

(by the chain rule)

so the finished derivative is

y '	=	8	( (1/2)*(ln(x))^-1/2 x^-1 )
	=	4*x^-1 (ln(x))^-1/2

additional explanation for the derivative of constant multiples
see another derivative of constant multiples example
practice gateway test
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Comments to Gavin LaRose
glarose@umich.edu
©2001 Gavin LaRose, University of Michigan Math Dept.