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 30E
To determine

To find: The derivative of the function f(x)=x2 at x=a.

Expert Solution

Answer to Problem 30E

The derivative of the function f(x) at x=a is 2a3_.

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)

Calculation:

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

Rewrite the function f(x) as follows.

f(x)=x2=1x2

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

f(a)=limh0f(a+h)f(a)h=limh0(1(a+h)2)(1a2)h=limh0(a2(a+h)2a2(a+h)2)h=limh0(a2(a2+h2+2ah))ha2(a+h)2

Perform the mathematical operations to simplify the numerator,

f(a)=limh0(a2a2h22ah)ha2(a+h)2=limh0(h22ah)ha2(a+h)2=limh0h(h+2a)ha2(a+h)2

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

f(a)=limh0(h+2a)a2(a+h)2=((0)+2a)a2(a+(0))2=2aa2a2=2a3

Thus, the derivative of the function f(x) at x=a is 2a3_.

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