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Study Guide for Chapter 2 Test  

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Study Guide for Chapter 2 Test  
Study Guide for Chapter 2 Test
Product Rule:
d
u v 
dx
Quotient Rule:
d u
 
dx  v 
Chain Rule:
d
 f g  ( x) 
dx
1. Use the definition of a derivative lim
f ( x  h)  f ( x )
dy
to find
if y = 2x2 – 1
h
dx
2. Use the definition of a derivative lim
2
f ( x  h)  f ( x )
to find f '( x) if f ( x) 
x 3
h
3. Use the definition of a derivative lim
f ( x  h)  f ( x )
dy
to find
if y = 3 2 x  5
h
dx
h 0
h 0
h 0
4. Use your answer from #3 to find the equation for the tangent line to the function at the point (7, 9).
1
 x5 ,x  0
5. Use the function f ( x)   4
to determine if the function is continuous and/or
,
x

0
2
 x  5

differentiable at x = 5.
1
 x5 ,x  0
6. Use the function f ( x)   4
to determine if the function is continuous and/or
 x 2  5 , x  0

differentiable at x = 0.
1
 x5 ,x  0
7. Find the equation for the derivative function of f ( x)   4
 x 2  5 , x  0

x2
3
1
8. Find the derivative of f ( x)  7 x  5 x   9 x  44 
 6
8
3
4
x
4 x
4
3
9. Find the derivative of y = 2 2 x 2  5 x  1
10. Find the derivative of y = 
1
x3
11. Find the derivative of f ( x) 
6x2  x
2x 1
12. Find the derivative of y =
( x  6)3
2 x2
13. Find the derivative of f ( x)  ( x  1)( x 2  x)
14. Find the derivative of f ( x)  2 x( x 2  x)3
15. Find the derivative of f ( x)  ( x  3) 4 (2 x 2  5) 2
16. Find the derivative of f ( x) 
( x  3)( x  5)
9 x2
17. Find the derivative of f ( x) 
( x  3)( x  5)
9 x 2 ( x  1)
18. Find the derivative of y  3x 2 x  1
19. Use the given information to answer the following questions.
x
–3
–2
–1
0
1
2
3
f (x) g(x) f '(x) g'(x)
4
0
-7
-2
4
2
7
-7
-2
4
-7
0
2
4
0
-2
0
2
2
0
4
4
7
2
-7
2
4
0
a. ( f  g )(0)
b. (3 f  4 g ) '(3)
c. ( fg ) '(1)
d. ( gf ) '(2)
 f 
e.   '(0)
g
 g2 
f.   '(2)
 f 
g. Find m '(1) if m( x) = g ( f ( x))
i. Find m '(2) if m( x)  g ( x) 2 f ( x)
h. Find m '(3) if m( x)  f ( x) g ( x)
20. Use implicit differentiation to find
dy
and the equations for the tangent and normal lines at the
dx
given point.
a. 4 x 2  8 x  9 y 2  18 y  5  36
b. xy 2  2 x3  y 2  10
at the point
 11 
 4, 
 3
at the points at which x=1.
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