Derivatives of Quotients

f '(x) g(x)

f(x) g '(x)

Example:

f(x) = (tan(x) - 1) / e^x

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

f(x)	=	tan(x) - 1

		e^x

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,

f(x)	=	tan(x) - 1

		e^x

we can think of as

f(x)	=	g(x)

		h(x)

So the derivative is

f(x)'	= (	g(x)	)'

		h(x)

=	g '(x)	h(x)	-	g(x)	h '(x)

	( h(x) )²
=	( tan(x) - 1 )'	( e^x )	-	( tan(x) - 1 )	( e^x )'

	( e^x )²

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

( tan(x) - 1 )'	=	1 / (cos(x))² - 0	(by the derivative rule for sums, derivative rules for basic functions, and the derivative rule for constants)
( e^x )'	=	e^x	(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.