Ma 113 Functions And Inverse Functions, The Exponential Function And The Logarithm Worksheet Page 18
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41Worksheet # 13: Chain Rule
1. (a) Carefully state the chain rule using complete sentences.
1
(b) Suppose
and are differentiable functions so that (2) = 3,
(2) =
1, (2) =
, and (2) = 2.
4
Find each of the following:
2
i.
(2) where ( ) =
[ ( )]
+ 7.
3
ii.
(2) where ( ) = (
( )).
2
2. Suppose that ( ) =
sin
( ) + 4. Find three functions , , and
so that ( ) = ( ( ( ))).
3
3. Given the following functions: ( ) = sec( ), and ( ) =
2 + 1. Find:
(a)
( ( ))
(d)
( ( ))
(b)
( )
(c)
( )
(e) (
) ( )
4. Differentiate each of the following and simplify your answer.
(3
+ )
3
(a)
( ) =
2
+ 7 + 3
(d)
( ) =
(b) ( ) = tan(sin( ))
2
2
(c)
( ) = sec
( ) + tan
( )
(e) ( ) = sin(sin(sin( )))
5. Find an equation of the tangent line to the curve at the given point.
2 3
(a)
( ) =
,
= 2
2
(b)
( ) = sin( ) + sin
( ),
= 0
6. Compute the derivative of
in two ways:
+1
(a) Using the quotient rule.
2
1
(b) Rewrite the function
= (
+ 1)
and use the product and chain rule.
+1
Check that both answers give the same result.
7. If ( ) =
4 + 3 ( ) where (1) = 7 and
(1) = 4, find
(1).
8. Let ( ) =
( ) and ( ) =
( ) where some values of
and
are given by the table
( )
( )
( )
( )
-1
4
4
-1
-1
2
3
4
3
-1
3
-1
-1
3
-1
4
3
2
2
-1
Find:
( 1)
(3) and
(2).
2
9. Find all
values so that ( ) = 2 sin( ) + sin
( ) has a horizontal tangent at .
10. Suppose that at the instant when the radius of a circle of a circle is 5 cm, the radius is decreasing at
a rate of 3 cm/sec. Find the rate of change of the area of the circle when the the radius is 5 cm.
2
2
11. Differentiate both sides of the double-angle formula for the cosine function, cos(2 ) = cos
( ) sin
( ).
Do you obtain a familiar identity?
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