#### To determine

**To find:** The derivative of the function f(x)=x+1x.

#### Answer

The derivative of f(x) is f′(x)= 1−1x2.

#### Explanation

**Given:**

The function is f(x)=x+1x.

**Formula used:**

The derivative of a function *f ,* denoted by f′(x), is

f′(x)=limh→0f(x+h)−f(x)h (1)

**Calculation:**

Obtain the derivative of the function f(x).

Compute f′(x) by using the equation (1),

f′(x)=limh→0f(x+h)−f(x)h=limh→0(x+h)+1(x+h)−(x+1x)h

Expand the numerator,

f′(x)=limh→0(x+h)2+1x+h−{x2+1x}h

Simplify the numerator,

f′(x)=limh→0x((x+h)2+1)−(x+h)(x2+1)(x+h)hx=limh→0x(x2+h2+2xh+1)−(x3+hx2+x+h)(x+h)hx=limh→0x3+xh2+2x2h+x−x3−hx2−x−h(x+h)hx=limh→0xh2+x2h−h(x+h)hx

=limh→0h(xh+x2−1)(x+h)hx

Since the limit *h* approaches zero but not equal to zero, cancel the common term *h* from both the numerator and the denominator,

f′(x) = limh→0(xh+x2−1)(x+h)x =x(0)+x2−1(x+0)x =x2−1x2 =1−1x2

Thus the value of the derivative is f′(x)=1−1x2.

#### To determine

**To Check:** The answer to part (a) is reasonable by comparing graphs f and f′.

#### Answer

The answer to part (a) is reasonable.

#### Explanation

**Graph:**

Use the online graphing calculator to draw the graph of f(x) and f′(x) as shown below in Figure 1.

**Observation:**

From Figure 1, it is observed that the function f(x)=x+1x has a horizontal tangent at x=−1 and x=1 and f′(x)=0 at x=−1 and x=1. That is,

When x=−1,

f′(x)=1−1(−1)2=1−1=0

When x=1,

f′(x)=1−1(1)2=1−1=0

Thus, the derivative f′(x)=0 when *f* has a horizontal tangent at x=−1 and x=1.

Also, the value of f′(x)>0 when slope of f(x) has positive value and f′(x)<0 when slope of f(x) has negative value.

Thus, it can be concluded that the answer to part (a) is reasonable.