# The derivative of the function f ( t ) = 2 t 3 + t at t = a .

### Single Variable Calculus: Concepts...

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

### Single Variable Calculus: Concepts...

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

#### Solutions

Chapter 2.6, Problem 28E
To determine

## To find: The derivative of the function f(t)=2t3+t at t=a.

Expert Solution

The derivative of the function f(t) at t=a is 6a2+1_.

### 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(t) at t=a.

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

f(a)=limh0f(a+h)f(a)h=limh0(2(a+h)3+(a+h))(2(a)3+(a))h=limh0(2(a3+3a2h+3ah2+h3)+(a+h))(2a3+a)h=limh0(2a3+6a2h+6ah2+3h3+(a+h))(2a3+a)h

Use elementary algebra to simplify the numerator as follows,

f(a)=limh02a3+6a2h+6ah2+3h3+a+h2a3ah=limh0(2a32a3)+(6a2h+6ah2+3h3+h)+(aa)h=limh0h(6a2+6ah+3h2+1)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(6a2+6ah+3h2+1)=(6a2+6a(0)+3(0)2+1)=6a2+1

Thus, the derivative of the function f(t) at t=a is 6a2+1.

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