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 2.6, Problem 31E
To determine

To find: The derivative of the function f(x)=12x at x=a.

Expert Solution

Answer to Problem 31E

The derivative of the function f(x) at x=a is 112a_.

Explanation of Solution

Formula used:

The derivative of a function f at a number a, denoted by f(a) is,

f(a)=limh0f(a+h)f(a)h (1)

Difference of squares formula: (a2b2)=(a+b)(ab).

Calculation:

Obtain the derivative of the function f(x) at x=a.

Use the equation (1) to compute f(a),

f(a)=limh0f(a+h)f(a)h=limh0(12(a+h))(12a)h

Multiply both the numerator and the denominator by the conjugate of the numerator,

f(a)=limh0(12(a+h))(12a)h×(12(a+h))+(12a)(12(a+h))+(12a)=limh0((12(a+h))(12a))×((12(a+h))+(12a))h((12(a+h))+(12a))

Apply the difference of square formula,

f(a)=limh0(12(a+h))2(12a)2h(12(a+h)+12a)=limh0(12(a+h))(12a)h(12(a+h)+12a)=limh012a2h1+2ah(12(a+h)+12a)=limh02hh(12(a+h)+12a)

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

f(h)=limh02(12(a+h)+12a)=2(12(a+0)+12a)=2(12a+12a)=2212a

=112a

Thus, the derivative of the function f(x) at x=a is 112a_.

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