Derivatives of Quotients

f '(x) g(x)

f(x) g '(x)

Example:

G = (8*t - (e)⁵) / 5^t

The first thing to notice when finding the derivative of this function is that it is a quotient, as shown below:

G	=	8*t - (e)⁵

		5^t

The Derivative Rule for Quotients:

The derivative of a quotient is the derivative of the numerator times the denominator minus the numerator times the derivative of the denominator, all divided by the denominator squared.
If

z	= (	f(x)	)

		g(x)

then the derivative of z is

So our example,

G	=	8*t - (e)⁵

		5^t

we can think of as

G	=	f(t)

		g(t)

So the derivative is

G'	= (	f(t)	)'

		g(t)

=	f '(t)	g(t)	-	f(t)	g '(t)

	( g(t) )²
=	( 8*t - (e)⁵ )'	( 5^t )	-	( 8*t - (e)⁵ )	( 5^t )'

	( 5^t )²

and we just need to know each of the derivatives on the right-hand side of the equation. In this case these are

( 8*t - (e)⁵ )'	=	8 - 0	(by the derivative rule for sums, rule for constant multiples, and the derivative rule for constants)
( 5^t )'	=	(ln(5))*5^t	(by the derivative rules for basic functions)

so the finished derivative is

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