(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* = -2*x* - 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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Sun Aug 3 18:53:29 PDT 1997