Kuta Software - Infinite Calculus
Name___________________________________
Differentiation - Product Rule
Date________________ Period____
Differentiate each function with respect to x.
1) y = − x 3 (3 x 4 − 2)
2) f ( x) = x 2 (−3 x 2 − 2)
3) y = (−2 x 4 − 3)(−2 x 2 + 1)
4) f ( x) = (2 x 4 − 3)( x 2 + 1)
5) f ( x) = (5 x 5 + 5)(−2 x 5 − 3)
6) f ( x) = (−3 + x −3 )(−4 x 3 + 3)
7) y = (−2 x 4 + 5 x 2 + 4)(−3 x 2 + 2)
8) y = ( x 4 + 3)(−4 x 5 + 5 x 4 + 5)
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Worksheet by Kuta Software LLC
9) y = (5 x 4 − 3 x 2 − 1)(−5 x 2 + 3)
(
10) f ( x) = −10 x 2 − 7
11) y = (5 + 3 x
−2
5
2
)
x + 9 (2 x 3 + 4)
)(4 x 5 + 6 x 3 + 10)
12) y = (−6 x 4 + 2 + 6 x −4 )(6 x 4 + 7)
13) f ( x) = (−7 x + 10 x + 8)( x + 10)
4
2
5
2
Critical thinking question:
14) A classmate claims that ( f ⋅ g)' = f ' ⋅ g' for any functions f and g. Show an example that proves your
classmate wrong.
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Worksheet by Kuta Software LLC
Kuta Software - Infinite Calculus
Name___________________________________
Differentiation - Product Rule
Date________________ Period____
Differentiate each function with respect to x.
1) y = − x 3 (3 x 4 − 2)
2) f ( x) = x 2 (−3 x 2 − 2)
f ' ( x) = x 2 ⋅ −6 x + (−3 x 2 − 2) ⋅ 2 x
= −12 x 3 − 4 x
dy
= − x 3 ⋅ 12 x 3 + (3 x 4 − 2) ⋅ −3 x 2
dx
= −21 x 6 + 6 x 2
3) y = (−2 x 4 − 3)(−2 x 2 + 1)
4) f ( x) = (2 x 4 − 3)( x 2 + 1)
f ' ( x) = (2 x 4 − 3) ⋅ 2 x + ( x 2 + 1) ⋅ 8 x 3
= 12 x 5 + 8 x 3 − 6 x
dy
= (−2 x 4 − 3) ⋅ −4 x + (−2 x 2 + 1) ⋅ −8 x 3
dx
= 24 x 5 − 8 x 3 + 12 x
5) f ( x) = (5 x 5 + 5)(−2 x 5 − 3)
6) f ( x) = (−3 + x −3 )(−4 x 3 + 3)
f ' ( x) = (5 x 5 + 5) ⋅ −10 x 4 + (−2 x 5 − 3) ⋅ 25 x 4
= −100 x 9 − 125 x 4
f ' ( x) = (−3 + x −3 ) ⋅ −12 x 2 + (−4 x 3 + 3) ⋅ −3 x −4
9
= 36 x 2 − 4
x
7) y = (−2 x 4 + 5 x 2 + 4)(−3 x 2 + 2)
dy
= (−2 x 4 + 5 x 2 + 4) ⋅ −6 x + (−3 x 2 + 2)(−8 x 3 + 10 x)
dx
= 36 x 5 − 76 x 3 − 4 x
8) y = ( x 4 + 3)(−4 x 5 + 5 x 4 + 5)
dy
= ( x 4 + 3)(−20 x 4 + 20 x 3 ) + (−4 x 5 + 5 x 4 + 5) ⋅ 4 x 3
dx
= −36 x 8 + 40 x 7 − 60 x 4 + 80 x 3
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Worksheet by Kuta Software LLC
9) y = (5 x 4 − 3 x 2 − 1)(−5 x 2 + 3)
dy
= (5 x 4 − 3 x 2 − 1) ⋅ −10 x + (−5 x 2 + 3)(20 x 3 − 6 x)
dx
= −150 x 5 + 120 x 3 − 8 x
(
10) f ( x) = −10 x 2 − 7
5
)
2
x + 9 (2 x 3 + 4)
14
f ' ( x) = (−10 x − 7 x + 9) ⋅ 6 x + (2 x + 4)(−20 x −
x
5
2
5
2
= −100 x 4 −
2
238 x
5
12
5
+ 54 x 2 − 80 x −
−2
3
5
)
56
5x
11) y = (5 + 3 x
−
3
3
5
)(4 x 5 + 6 x 3 + 10)
dy
= (5 + 3 x −2 )(20 x 4 + 18 x 2 ) + (4 x 5 + 6 x 3 + 10) ⋅ −6 x −3
dx
60
= 100 x 4 + 126 x 2 + 18 − 3
x
12) y = (−6 x 4 + 2 + 6 x −4 )(6 x 4 + 7)
dy
−4
−5
= (−6 x 4 + 2 + 6 x ) ⋅ 24 x 3 + (6 x 4 + 7)(−24 x 3 − 24 x )
dx
168
= −288 x 7 − 120 x 3 − 5
x
13) f ( x) = (−7 x + 10 x + 8)( x + 10)
f ' ( x) = (−7 x + 10 x + 8) ⋅ 2 x + ( x + 10)(−28 x + 4 x )
2
5
4
4
5
2
2
5
−
2
3
7
5
= −42 x − 280 x + 24 x + 16 x +
3
3
5
40
x
3
5
Critical thinking question:
14) A classmate claims that ( f ⋅ g)' = f ' ⋅ g' for any functions f and g. Show an example that proves your
classmate wrong.
Many answers. Ex: f = 2 x, g = 4, 8 ≠ 0
Create your own worksheets like this one with Infinite Calculus. Free trial available at KutaSoftware.com
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Worksheet by Kuta Software LLC