Derivatives of Constant Multiples

Example:

Q(z) = (1/2)*(z + p) e^{4*z + 5}

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:

Q(z)

1/2

(z + p) e^{4*z + 5}

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,

Q(z)

1/2

(z + p) e^{4*z + 5}

we can think of as

Q(z)

f(z)

So the derivative is

Q '(z)	= (	c	f(z)	)'
	=	c	f '(z)
	=	1/2	((z + p) e^{4*z + 5})'

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

(z + p) e^{4*z + 5}

(1 + 0) e^{4*z + 5} + (z + p) e^{4*z + 5} (4 + 0)

(by the product rule, and the derivative rule for sums, and chain rule)

so the finished derivative is

Q '(z)	=	1/2	( (1 + 0) e^{4z + 5} + (z + p) e^{4z + 5} (4 + 0) )
	=	(1/2)(e^{4z + 5} + 4(z + p) e^{4z + 5})

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.