BuyFind

Single Variable Calculus: Concepts...

4th Edition
James Stewart
Publisher: Cengage Learning
ISBN: 9781337687805
BuyFind

Single Variable Calculus: Concepts...

4th Edition
James Stewart
Publisher: Cengage Learning
ISBN: 9781337687805

Solutions

Chapter 3.1, Problem 59E
To determine

To show: If f(x)=1x, then f(x)=1x2.

Expert Solution

Explanation of Solution

Definition of derivative:

The derivative of a function f(x) at a number a, denoted by f(a), is defined as f(a)=limh0(f(a+h)f(a)h) if the limit exists.

Proof:

Use the definition of derivative and obtain the derivative of the function f(x) as follows.

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

Since f(x)=1x, f(x+h)=1x+h, the derivative of the function f(x) becomes,

f(x)=limh0(1x+h1xh)=limh0(xx(x+h)(x+h)x(x+h)h)=limh0(x(x+h)x(x+h)h)=limh0(hx(x+h)h1)

Perform the arithmetic operation and simplify the terms,

f(x)=limh0(hx(x+h)1h)=limh0(1x(x+h))=1x(x+0)=1x2

Therefore, the derivative of the function f(x)=1x is f(x)=1x2.

Hence, the required proof is obtained.

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