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*SOLUTION 1:* Differentiate . Apply the product rule.
Then

(Factor an *x* from each term.)

.

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

.

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

arcarcarcarc

arcarc

= ( arcarc .

*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 .

*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* .

*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.

*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.)

.

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

(Recall that .)

.

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

(Recall that .)

.

*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

.

*SOLUTION 11:* Differentiate arc . What conclusion can be drawn from your answer about function *y* ? What conclusion can be drawn about functions arc and ? First, differentiate, applying the chain rule to the inverse cotangent function. Then

= 0 .

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

for all admissable values of *x* ,

i.e.,

arc for all admissable values of *x* .

In particular, if *x* = 1 , then

arc

i.e.,

.

Thus, *c* = 0 and arc for all admissable values of *x* . We conclude that

arc .

Note that this final conclusion follows even more simply and directly from the definitions of these two inverse trigonometric functions.

*SOLUTION 12:* Differentiate . Begin by applying the product rule to the first summand and the chain rule to the second summand. Then

.

*SOLUTION 13:* Find an equation of the line tangent to the graph of
at *x*=2 . If *x* = 2 , then , so that the line passes through the point . The slope of the tangent line follows from the derivative

(Recall that when dividing by a fraction, one must invert and multiply by the reciprocal. That is .)

.

The slope of the line tangent to the graph at *x* = 2 is

.

Thus, an equation of the tangent line is

or

or

.

*SOLUTION 14:* Evaluate . Since and , it follows that
takes the indeterminate form . Thus, we can apply
L'Hpital's Rule. Begin by differentiating the numerator and denominator separately. DO NOT apply the quotient rule ! Then

=

=

(Recall that when dividing by a fraction, one must invert and multiply by the reciprocal. That is .)

=

=

= .

*SOLUTION 15:* A movie screen on the front wall in your classroom is 16 feet high and positioned 9 feet above your eye-level. How far away from the front of the room should you sit in order to have the ``best" view ? Begin by introducing variables *x* and . (See the diagram below.)

From trigonometry it follows that

,

so that

.

In addition,

so that

.

It follows that

,

that is, angle is explicitly a function of distance *x* . Now find the value of *x* which maximizes the value of function . Begin by differentiating function and setting the derivative equal to zero. Then

.

.

Now solve this equation for *x* . Then

iff

iff

iff

iff

iff

feet .

Use the first or second derivative test (The first derivative test is easier.) to verify that this value of *x*
determines a maximum value for .

Tue Sep 16 16:10:59 PDT 1997