   Chapter 2.6, Problem 21E

Chapter
Section
Textbook Problem

In Exercises 17-22, find the slope of the tangent line to the graph of the function at the given point, and determine an equation of the tangent line.21. f ( x ) = − 1 x at (3, − 1 3 )

To determine
The slope of the tangent line to the graph of the given function and equation of the tangent.

Explanation

Given information:

The given function is f(x)=1x and the given point is (3,13) .

Calculation:

The given function is,

f(x)=1x (1)

The four steps to compute the value of f(x) are shown below,

Step 1: Substitute x+h for x in equation (1), to find the value of f(x+h) .

f(x+h)=1x+h (2)

Step 2: Subtract equation (1) from equation (2),

f(x+h)f(x)=1x+h(1x)=1x+h+1x=xx(x+h)+x+hx(x+h) (Commondenominatorx(x+h))

Simplify the above expression,

f(x+h)f(x)=x+x+hx(x+h)=hx(x+h)

Step 3: Divide the above equation by h to form the quotient,

f(x+h)f(x)h=1h(hx(x+h))=1x(x+h)

Step 4: Compute the value of f(x) ,

f(x)=limh0f(x+h)f(x)h=limh0(1

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