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 27E
To determine

To find: The derivative of the function f(x)=3x24x+1 at x=a.

Expert Solution

Answer to Problem 27E

The derivative of the function f(x) at x=a is 6a4.

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.

Compute f(a) by using the equation (1).

f(a)=limh0f(a+h)f(a)h=limh0(3(a+h)24(a+h)+1)(3a24a+1)h=limh0(3(a2+h2+2ah)4a4h+1)(3a24a+1)h=limh03a2+3h2+6ah4a4h+13a2+4a1h

Use elementary algebra to simplify the numerator as follows,

f(a)=limh0(3a23a2)+3h2+6ah+(4a4a)4h+(11)h=limh03h2+6ah4hh=limh0h(3h+6a4)h

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(3h+6a4)=(3(0)+6a4)=6a4

Thus, the derivative of the function f(x) at x=a is 6a4.

Have a homework question?

Subscribe to bartleby learn! Ask subject matter experts 30 homework questions each month. Plus, you’ll have access to millions of step-by-step textbook answers!