Download Quotient Rule Practice

Quotient Rule Practice
Find the derivatives of the following rational functions.
a)
x2
x+1
b)
x4 + 1
x2
c)
sin(x)
x
Solution
a)
x2
x+1
The quotient rule tells us that if u(x) and v(x) are differentiable functions,
and v(x) is non-zero, then:
0
u(x)
u0 (x) v(x) − u(x) v 0 (x)
=
v(x)
(v(x))2
In this problem u = x2 and v = x + 1, so u0 = 2x and v 0 = 1. Applying the
quotient rule, we see that:
2 0
x
2x · (x + 1) − x2 · 1
=
x+1
(x + 1)2
2x2 + 2x − x2
=
(x + 1)2
2
x + 2x
=
.
(x + 1)2
b)
x4 + 1
x2
Here u(x) = x4 + 1, u0 (x) = 4x3 , v(x) = x2 and v 0 (x) = 2x. The quotient
rule tells us that:
4
0
x +1
4x3 · x2 − (x4 + 1) · 2x
=
2
x
(x2 )2
4x5 − 2x5 − 2x
=
x4
4
2x − 2
=
.
x3
1
c)
sin(x)
x
The derivative of sin(x) is cos(x) and the derivative of x is 1, so the quotient
rule tells us that:
sin(x)
x
0
=
=
(cos(x)) · x − (sin(x)) · 1
x2
x cos(x) − sin(x)
.
x2
When we learn to evaluate this expression at x = 0, it will tell us that the
slope of the graph of sinc(x) = sinx x is 0 when x = 0.
2
MIT OpenCourseWare
http://ocw.mit.edu
18.01SC Single Variable Calculus
Fall 2010
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