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 3.3, Problem 25E

(a)

To determine

To find: The derivative of the function f(x)=secxx.

Expert Solution

Answer to Problem 25E

The derivative of f(x) is f(x)=secxtanx1_.

Explanation of Solution

Given:

The function is f(x)=secxx.

Derivative rules:

(1) Difference Rule: ddx[f(x)g(x)]=ddx[f(x)]ddx[g(x)]

(2) Power Rule: ddx(xn)=nxn1

Calculation:

Obtain the derivative of f(x).

f(x)=ddx(f(x)) =ddx(secxx) 

Apply the difference rule (1),

f(x)=ddx[secx]ddx[x]=secxtanxddx[x]

Apply the power rule (2) and simplify further,

f(x)=secxtanx[1x11]=secxtanx1

Therefore, the derivative of f(x) is f(x)=secxtanx1_.

(b)

To determine

To check: The derivatives of f(x) obtained in part (a) is reasonable or not by using the graphs of f(x) and f(x).

Expert Solution

Answer to Problem 25E

The derivatives of f(x) is reasonable.

Explanation of Solution

Graph:

Use the online graphing calculator and draw the graph of f(x)=secxx and f(x)=secxtanx1 as shown below in Figure 1.

Single Variable Calculus: Concepts and Contexts, Enhanced Edition, Chapter 3.3, Problem 25E

Observation:

From Figure 1, it is noticed that,

If f(x) is positive then f(x) is increasing function.

If f(x) is negative then f(x) is decreasing function.

If f(x) crosses the x axis (f(x)=0), then f(x) is local extrema (that is, local minimum or local maximum).

Therefore, it can be concluded that the derivative of the function f(x)=secxx is reasonable.

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