Calculus 1
Worksheet #12
Package Rule
Learn: Package Rule
Example: f ( x)  4(1  7 x)9  f '( x)  (9)(4)(1  7 x)91 (7)  252(1  7 x)8
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1.At the specified point, find the equation of the normal to the curve: f(x) = 100  x2
if domain = x  10;
Point = (−6, __ )
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2. For what value of k , for g (x) =
1
x + k , is g a normal to f(x) = x2 +1 (hint: you know the
4
slope!)
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Directions: find f ‘(x) for problems 3−16
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3. f ( x)  (3x  2)8
4. f ( x)  (1  x)6
5. f ( x)  (1  2 x 2 )3
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7. f ( x) 
6. f ( x)  (1  x 2 )5
1
( x  2)3
8. f ( x)  x  1
2
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9. f ( x)  (2 x3  3x  1)4
10. f ( x)  x 2  2 x  1
11. f ( x) 
3
1
x3
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1
12. f ( x)  ( x 2  4) 2
13. f ( x)  6 x 4  8x3
14. f ( x)  x 2 ( x 2  3)
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15. f ( x)  ( x3  2 x)(3x 2 )
16. f ( x) 
1
x5
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Write the equation of the normal at the given point.
17. f ( x)  2 x3  2 x  7 at [−1, f(−1)]
18. f ( x)  (2 x  3)3 at [0, f(0)]
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19. f ( x)  x  1 at [5, f(5)]
20. f ( x) 
2
at [2, f(2)]
x3
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Revised: 7/28/2017
Calculus 1
Worksheet #12
Package Rule
Answers:
4
 x  6
3
5. 12 x(1  2 x 2 )2
1. y  8  
1
2
6. 10 x(1  x 2 ) 4
2. k  5
( x  1)
9.
4(2 x3  3x  1)3  6 x 2  3
10.
13. 24 x2  x 1
14. 2 x(2 x 2  3)
17.
y7 
1
 x  1
4
Revised: 7/28/2017
x2  2x 1
18. y 
1
x  27
54
3. 24(3x  2)7
6 x
( x  2) 4
1
11.
3 3 ( x  3) 4
7.
2
15. 3x 2 (5 x 2  6)
19. y  4 x  22
4. 6(1  x)5
8.
12.
1
2 x 1
x
3
( x 2  4) 2
5
16. 6
x
8
61
20. y  x 
3
12