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Chain Rule - Kuta Software
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Kuta Software - Infinite Calculus
Name___________________________________
Differentiation - Chain Rule
Date________________ Period____
Differentiate each function with respect to x. 1) y = ( x 3 + 3)
5
3) y = (−5 x 3 − 3)
2) y = (−3 x 5 + 1)
3
4) y = (5 x 2 + 3)
4
5) f ( x) =
4
−3 x 4 − 2
6) f ( x) =
7) f ( x) =
3
−2 x 4 + 5
8) y = (− x 4 − 3)
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-1-
3
−2 x 2 + 1
−2
Worksheet by Kuta Software LLC
9) y = (3 x 3 + 1)(−4 x 2 − 3) 4
10) y =
( x 3 + 4) 5 3x4 − 2
11) y = (( x + 5) − 1) 5
12) y = (5 x 3 − 3) 5
4
4
−4 x 5 − 3
Critical thinking question: 13) Give a function that requires three applications of the chain rule to differentiate. Then differentiate the function.
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-2-
Worksheet by Kuta Software LLC
Kuta Software - Infinite Calculus
Name___________________________________
Differentiation - Chain Rule
Date________________ Period____
Differentiate each function with respect to x. 5
1) y = ( x 3 + 3) dy 4 = 5( x 3 + 3) ⋅ 3 x 2 dx 4 = 15 x 2 ( x 3 + 3)
3) y = (−5 x 3 − 3)
2) y = (−3 x 5 + 1)
dy 2 = 3(−3 x 5 + 1) ⋅ −15 x 4 dx 2 = −45 x 4 (−3 x 5 + 1)
3
4) y = (5 x 2 + 3)
dy = 3(−5 x 3 − 3) 2 ⋅ −15 x 2 dx 2 = −45 x 2 (−5 x 3 − 3)
5) f ( x) =
4
3
7) f ( x) =
3
− 2)
− 1 f ' ( x) = (−2 x 2 + 1) 2 ⋅ −4 x 2 2x =−
3 4
(−2 x
−2 x 4 + 5
8) y = (− x 4 − 3) 2
3(−2 x + 5)
2 3
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2
+ 1)
1 2
−2
dy = −2(− x 4 − 3) −3 ⋅ −4 x 3 dx 8x3 = (− x 4 − 3) 3
− 1 f ' ( x) = (−2 x 4 + 5) 3 ⋅ −8 x 3 3 8x3 =− 4
−2 x 2 + 1 1
− 1 f ' ( x) = (−3 x 4 − 2) 4 ⋅ −12 x 3 4 3x3 =−
(−3 x
4
dy = 4(5 x 2 + 3) 3 ⋅ 10 x dx 3 = 40 x(5 x 2 + 3)
6) f ( x) =
−3 x 4 − 2
4
3
-1-
Worksheet by Kuta Software LLC
9) y = (3 x 3 + 1)(−4 x 2 − 3) 4 dy 3 4 = (3 x 3 + 1) ⋅ 4(−4 x 2 − 3) ⋅ −8 x + (−4 x 2 − 3) ⋅ 9 x 2 dx 3 = x(−4 x 2 − 3) (−132 x 3 − 32 − 27 x)
10) y =
( x 3 + 4) 5 3x4 − 2 4
5
dy (3 x 4 − 2) ⋅ 5( x 3 + 4) ⋅ 3 x 2 − ( x 3 + 4) ⋅ 12 x 3 = dx (3 x 4 − 2) 2 4 3 x 2 ( x 3 + 4) (11 x 4 − 10 − 16 x) = (3 x 4 − 2) 2 11) y = (( x + 5) − 1) 5
4
3 dy 5 4 = 4(( x + 5) − 1) ⋅ 5( x + 5) dx 3
= 20(( x + 5) − 1) ⋅ ( x + 5) 5
12) y = (5 x 3 − 3) 5
4
4
−4 x 5 − 3 3
1
− 1 dy = (5 x 3 − 3) 5 ⋅ (−4 x 5 − 3) 4 ⋅ −20 x 4 + (−4 x 5 − 3) 4 ⋅ 5(5 x 3 − 3) 4 ⋅ 15 x 2 dx 4 2 3 5 x (5 x − 3) 4 (−65 x 5 − 45 + 3 x 2 ) =
(−4 x
5
− 3)
3 4
Critical thinking question: 13) Give a function that requires three applications of the chain rule to differentiate. Then differentiate the function.
(
6
)
7
Many answers: Ex y = ((2 x + 1) + 2) + 3 6 6 5 dy 5 5 4 = 7 ((2 x + 1) + 2) + 3 ⋅ 6((2 x + 1) + 2) ⋅ 5(2 x + 1) ⋅ 2 dx
(
5
)
Create your own worksheets like this one with Infinite Calculus. Free trial available at KutaSoftware.com
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-2-
Worksheet by Kuta Software LLC
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