Derivatives of Constant Multiples

Example:

V = (1/3)*e^{y² + 3}

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:

1/3

e^{y² + 3}

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,

1/3

e^{y² + 3}

we can think of as

f(y)

So the derivative is

V '	= (	c	f(y)	)'
	=	c	f '(y)
	=	1/3	(e^{y² + 3})'

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

e^{y² + 3}

e^{y² + 3} (2*y + 0)

(by the chain rule)

so the finished derivative is

V '	=	1/3	( e^{y² + 3} (2*y + 0) )
	=	(2/3)*y e^{y² + 3}

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.