### Product and Quotient Rule

```Section 2.3: Product and
Quotient Rule
Objective:
Students will be able to use the product
and quotient rule to take the derivative of
differentiable equations
Review: Definition of Derivative
The derivative of f at x is given by
f ( x  x)  f ( x)
f ( x) xlim
0
x
Provided the limit exists. For all x for which this limit
exists, f’ is a function of x.
Product Rule

Theorem 2.7:The product of two differentiable function f
and g is itself differentiable. Moreover, the derivative of fg is
the first function times the derivative of the second , plus
the second function times the derivative of the first.
d
 f ( x) g ( x)  f ( x) g ( x)  g ( x) f ( x)
dx

You can reverse the order in which you take the derivative
of the terms in the product rule.
Example #1
(5x  4 x )(6  2 x)
2
Step 1:
(5 x  4 x 2 )
d
d
[6  2 x]  (6  2 x) [5 x  4 x 2 ]
dx
dx
(derivative of the second term)(first term)+(derivative of the first term)(second term)
Step 2:
[(5x  4x )(2)]  [(6  2x)(5  8x)]
2
-Take derivative
Step 3:
10x  8x 2  30  48x  10x  16x 2
-Simplify
Step 4:
 24x  28x  30
2
-Simplify
Example #2
x (5x  2x)
2
Step 1:
x (10x  2)  (5x  2x)(.5x
2
(1/ 2)
-Take derivative
Step 2:
10x
( 3 / 2)
 2 x  (5 / 2) x
-Simplify
Step 3:
12.5x
(3 / 2)
 3x
-Simplify
(1/ 2 )
( 3 / 2)
x
(1/ 2)
)
Example #3
( x  3x  1)(3x  2)
3
Find the tangent line at point (-2,1) using the above equation
Step 1:
[(x3  3x  1)(3)]  [(3x 2  3)(3x  2)]
-Take derivative
Step 2:
Step 3:
Step 4:
Step 5:
3x3  9 x-Simplify
 3  9 x3  6 x 2  9 x  6
12x3  6 x 2  18x  3
-Simplify
-plug in x=-2 from the point
12(2)3  6(2)2 18(2)  3 (-2,1)
to get the slope of the
 96  24  36  3  39
tangent line
-Simplify
Step 6:
y  1  39( x  2)
-Plug slope & point into the point slope equation
Quotient Rule

Theorem 2.8:The quotient f/g of two differentiable
functions f and g is itself differentiable at all values of
x for which g(x)≠ 0. Moreover, the derivative of f/g is
given by the denominator times the derivative of the
numerator minus the numerator times the derivative
of the denominator, all divided by the square of the
denominator.
d  f ( x)  g ( x) f ( x)  f ( x) g ( x)

,


2
dx  g ( x) 
g ( x)

g ( x)  0
You can not reverse the order in which you take the
derivative of the terms in the quotient rule.
Example #1
4x  2
y 2
x 6
d
d 2
( x  6) [4 x  2]  (4 x  2) [ x  6]
d 4x  2
dx
dx
[ 2
]
dx x  6
( x 2  6) 2
2
Step 1:
(derivative of the top term)(bottom term)-(derivative of the bottom term)(top term)
(bottom term)2
Step 2:
Step 3:
[(x 2  6)(4)]  [(4 x  2)(2 x)]
( x 2  6) 2
 4 x 2  4 x  24
x 4  12x 2  36
-Simplify
-Take derivative
Example #2
5  (1 / x )
y 
x3  7
Step 1:
x[5  (1 / x)] -Get rid of fraction in the numerator by
x( x 3  7) multiply the numerator and denominator by x
Step 2:
5x 1
x4  7x
Step 3:
Step 4:
-Simplify
( x 4  7 x)(5)  (5x  1)(4 x 3  7)
( x 4  7 x) 2
 15x 4  4 x 3  7
x 8  35x 5  49x 2
-Simplify
-Take derivative
Example #3
x2  6
y 4
x 8
Find tangent equation at point (-1,3)
Step 1:
2 x( x 4  8)  4 x 3 ( x 2  6) -Take derivative
( x 4  8) 2
Step 2:
2 x 5  16x  4 x 5  24x 3 -Simplify
x 8  16x  64
Step 3:
Step 4:
 2 x 5  24x 3  16x
x 8  16x 4  64
-Simplify
 2(1)5  24(1)3  16(1) 10 -plug in x=1 from the

8
4
(1)  16(1)  64
81 point (-1,3) to find the
slope of the tangent line
Step 5:
y 3 
10
( x  1) -plug the slope and point into the point
81
slope formula
Combining the Product Rule & Quotient Rule
*For this type of problem use the quotient rule and with in the quotient rule use the
product rule to take the derivative of the numerator
Product rule for
(3 x  2)(4 x 2  8)
derivative of the
x9
numerator
2
2
Step 1: 3(4 x  8)  8 x(3x  2)( x  9)  (1)(3x  2)(4 x  8)
( x  9) 2
-Take derivative
2
2
3
2
Step 2: (12x  24  24x  16x)(x  9)  12x  24x  8 x  16
x 2  18x  81
-Simplify
3
3
2
2
2
3
2
Step 3: 12x  24x  24x  16x  108x  216 216x  144x  12x  24x  8 x  16
x 2  18x  81
-Simplify
3
2
Step 4: 24x  332x  144x  200
x 2  18x  81
-Simplify
```