.
The product rule is NOT necessary here.)
.
(Recall that .
The product rule is NOT necessary here.)
Then
.
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SOLUTION 2 : Differentiate 
. Apply the product rule.
Then
.
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SOLUTION 3 : Differentiate 
.
Apply the quotient rule.
Then
(Recall the well-known trigonometry identity 
.)
.
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SOLUTION 4 : Differentiate 
. Apply the product rule.
Then
.
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SOLUTION 5 : Differentiate 
. To avoid using the chain rule, first rewrite the problem as
.
Now apply the product rule. Then
.
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SOLUTION 6 : Differentiate 
. To avoid using the chain rule, recall the trigonometry identity 
, and first rewrite the problem as
.
Now apply the product rule twice. Then
(This is an acceptable answer. However, an alternative answer can be gotten by using the trigonometry identity
.)
.
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SOLUTION 7 : Differentiate 
. Rewrite g as a triple product and
apply the triple product rule. Then
so that the derivative is
.
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SOLUTION 8 : Evaluate 
. It may not be obvious, but this problem can be viewed as a differentiation problem. Recall that
.
If 
, then 
, and letting 
it follows that
.
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SOLUTION 9 : Differentiate 
. Apply the chain rule to both functions. (If necessary, review the section on the chain rule .) Then
(Recall that .)
.
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SOLUTION 10 : Differentiate 
. This is NOT a product of functions. It's a
composition of functions. Apply the chain rule. Then
.
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SOLUTION 11 : Differentiate 
.
Apply the quotient rule first, followed by the chain rule. Then
.
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SOLUTION 12 : Differentiate 
.
Apply the product rule first, followed by the chain rule. Then
.
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SOLUTION 13 : Differentiate 
. Apply the chain rule four times ! Then
.
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SOLUTION 14 : Differentiate 
.
Apply the quotient rule first. Then
(Apply the product rule in the first part of the numerator.)
.
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SOLUTION 15 : Find an equation of the line tangent to the graph of 
at x=-1 . If x= -1 then
so that
the tangent line passes through the point
(-1, 0 ) . The slope of the tangent line follows from the derivative
.
The slope of the line tangent to the graph at x = -1 is
= -2 .
Thus, an equation of the tangent line is
y - 0 = -2 (x - (-1) ) or y = -2x - 2 .
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SOLUTION 16 : Find an equation of the line perpendicular to the graph of 
at 
. If then
so that the tangent line passes through the point
. The slope of the tangent line follows from the derivative of y . Then
.
The slope of the line tangent to the graph at is
.
Thus, the slope of the line perpendicular to the graph at is
m = - 2 ,
so that an equation of the line perpendicular to the graph at is
or
.
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SOLUTION 17 : Assume that 
. Solve f'(x) = 0 for x in the interval 
.
Use the chain rule to find the derivative of f . Then
(It is a fact that if A B = 0 , then A=0 or B = 0 . )
so that
or 
.
If , then the only solutions x in are
or 
.
If , then the only solutions x in are
or 
.
Thus, the only solutions to f'(x) = 0 in the interval are
or .
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SOLUTION 18 : Use any method to verify that 
.
Then
(Apply the quotient rule.)
(Recall the well-known trigonometry identity .)
(Recall that 
.)
.
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