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4th Edition

James Stewart

Publisher: Cengage Learning

ISBN: 9781337687805

Chapter 2.7, Problem 47E

To determine

**To find:** The first, second, third and fourth derivatives of the function

Expert Solution

The first, second, third and fourth derivatives of the function is

**Given:**

The function is,

**Result Used:**

The derivative of a function *f* is denoted by

**Calculation:**

The first derivative of the function is computed as follows,

Simplified further to obtain the derivative of the function,

Therefore, the derivative of the function

The second derivative of the function is computed as follows,

Simplify the expression and obtain the derivatives,

Therefore, the second derivative of the function

The third derivative of the function is computed as follows,

Simplify the terms and obtain derivatives,

Therefore, the third derivative of the function

The fourth derivative of the function is computed as follows,

Thus, the fourth derivative of the function

**To sketch:** The graph of the functions

The graphs are consistent with the geometric interpretations of these derivatives.

**Graph:**

Use the online graphing calculator to draw the functions

**Geometrical Interpretation:**

Take several points on the domain and estimate slope of the tangent to the function

Thus, the derivative

The

Take several points on the domain and estimate slope of the tangent to the function

Therefore, the function

The

The straight line has same slope at all points. That is,

From the Figure 1, the estimated slope of the function

Therefore, the function