(Factor an *x* from each term.)

.

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* SOLUTION 2 :* Differentiate . Apply the quotient rule. Then

.

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* SOLUTION 3 :* Differentiate arcarc .
Apply the product rule. Then

arcarcarcarc

arcarc

= ( arcarc .

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* SOLUTION 4 :* Let arc . Solve *f*'(*x*) = 0 for *x* . Begin by differentiating *f* . Then

(Get a common denominator and subtract fractions.)

.

(It is a fact that if , then *A* = 0 .) Thus,

2(*x* - 2)(*x*+2) = 0 .

(It is a fact that if *AB* = 0 , then *A* = 0 or *B*=0 .) It follows that

*x*-2 = 0 or *x*+2 = 0 ,

that is, the only solutions to *f*'(*x*) = 0 are

*x* = 2 or *x* = -2 .

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* SOLUTION 5 :* Let . Show that *f*'(*x*) = 0 . Conclude that
. Begin by differentiating *f* . Then

.

If *f*'(*x*) = 0 for all admissable values of *x* , then *f* must be a constant function, i.e.,

for all admissable values of *x* ,

i.e.,

for all admissable values of *x* .

In particular, if *x* = 0 , then

i.e.,

.

Thus, and for all admissable values of *x* .

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* SOLUTION 6 :* Evaluate . It may not be obvious, but this problem can be viewed as a derivative problem. Recall that

(Since *h* approaches 0 from either side of 0, *h* can be either a positve or a negative number. In addition,
is equivalent to . This explains the following
equivalent variations in the limit definition of the derivative.)

.

If , then , and letting , it follows that

.

The following problems require use of the chain rule.

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* SOLUTION 7 :* Differentiate . Use the product rule first. Then

(Apply the chain rule in the first summand.)

(Factor out . Then get a common denominator and add.)

.

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* SOLUTION 8 :* Differentiate . Apply the chain rule twice. Then

(Recall that .)

.

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* SOLUTION 9 :* Differentiate . Apply the chain rule twice. Then

(Recall that .)

.

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* SOLUTION 10 :* Determine the equation of the line tangent to the graph of
at *x* = *e* . If *x* = *e* , then , so that the line passes through the point . The slope of the tangent line follows from the derivative (Apply the chain rule.)

.

The slope of the line tangent to the graph at *x* = *e* is

.

Thus, an equation of the tangent line is

.

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Tue Sep 16 16:10:59 PDT 1997