Q. 43 Use the Second Fundamental Theor... [FREE SOLUTION]

91影视

Chapter 4: Q. 43 (page 400)

Use the Second Fundamental Theorem of Calculus, if needed, to calculate each of the derivatives given below.
$\frac{d^{2}}{d x^{2}} {鈭�}_{x^{2}}^{1} 鈥� \ln 鈦� | t | d t$

Short Answer

Expert verified

Ans: $\frac{d^{2}}{d x^{2}} {鈭�}_{x^{2}}^{1} 鈥� \ln 鈦� | t | d t = 鈭� 2 \ln 鈦� (|x^{2}|) 鈭� 4$

Step by step solution

Step 1. Given information.

given, $\frac{d^{2}}{d x^{2}} {鈭�}_{x^{2}}^{1} 鈥� \ln 鈦� | t | d t$

Step 2. The objective is to calculate the derivative.

Now, if $f$ is continuous on role="math" localid="1648715451199" $[a, b]$ then for allrole="math" localid="1648715456581" $x 鈭� [a, b]$ .

role="math" $\frac{d}{d x} {鈭�}_{a}^{u (x)} 鈥� f (t) d t = f (u (x)) u^{鈥�} (x)$

The derivative can be written as,

role="math" localid="1648715535214" $\begin{aligned} \frac{d^{2}}{d x^{2}} {鈭�}_{x^{2}}^{1} 鈥� \ln 鈦� | t | d t \\ = \frac{d^{2}}{d x^{2}} 鈭� {鈭�}_{1}^{x^{2}} 鈥� \ln 鈦� | t | d t \end{aligned}$

So,

$\begin{aligned} f (u (t)) = \ln 鈦� | t | \\ f (u (x)) = \ln 鈦� (|x^{2}|) \\ u^{鈥�} (x) = 2 x \\ f (u (x)) u^{鈥�} (x) = 2 x \ln 鈦� (|x^{2}|) \end{aligned}$

Step 3. The derivative expression can be written as,

$\begin{aligned} \frac{d^{2}}{d x^{2}} {鈭�}_{x^{2}}^{1} 鈥� \ln 鈦� | t | d t \\ = \frac{d^{2}}{d x^{2}} 鈭� {鈭�}_{1}^{x^{2}} 鈥� \ln 鈦� | t | d t \\ = \frac{d}{d x} (\frac{d}{d x} 鈭� {鈭�}_{1}^{x^{2}} 鈥� \ln 鈦� | t | d t) \\ = \frac{d}{d x} (鈭� 2 x \ln 鈦� (|x^{2}|)) \\ = 鈭� 2 \frac{d}{d x} (x \ln 鈦� (|x^{2}|)) \\ = 鈭� 2 (\ln 鈦� (|x^{2}|) + x (2 x) \frac{1}{x^{2}}) \\ = 鈭� 2 (\ln 鈦� (|x^{2}|) + 2 x^{2} \frac{1}{x^{2}}) \\ = 鈭� 2 \ln 鈦� (|x^{2}|) 鈭� 2 (2) \\ = 鈭� 2 \ln 鈦� (|x^{2}|) 鈭� 4 \end{aligned}$

Therefore, the value islocalid="1648715687817" $鈭� 2 \ln 鈦� (|x^{2}|) 鈭� 4$

Unlock Step-by-Step Solutions & Ace Your Exams!

Full Textbook Solutions
Get detailed explanations and key concepts
Unlimited Al creation
Al flashcards, explanations, exams and more...
Ads-free access
To over 500 millions flashcards
Money-back guarantee
We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Recommended explanations on Math Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

91影视

Use the Second Fundamental Theorem of Calculus, if needed, to calculate each of the derivatives given below.
$\frac{d^{2}}{d x^{2}} {鈭�}_{x^{2}}^{1} 鈥� \ln 鈦� | t | d t$

Short Answer

Step by step solution

Step 1. Given information.

Step 2. The objective is to calculate the derivative.

Step 3. The derivative expression can be written as,

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Statistics

Pure Maths

Theoretical and Mathematical Physics

Mechanics Maths

Geometry

Applied Mathematics

Study anywhere. Anytime. Across all devices.

91影视

Use the Second Fundamental Theorem of Calculus, if needed, to calculate each of the derivatives given below. d2dx2鈭�x21鈥�ln鈦�|t|dt

Short Answer

Step by step solution

Step 1. Given information.

Step 2. The objective is to calculate the derivative.

Step 3. The derivative expression can be written as,

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Statistics

Pure Maths

Theoretical and Mathematical Physics

Mechanics Maths

Geometry

Applied Mathematics

Study anywhere. Anytime. Across all devices.

Use the Second Fundamental Theorem of Calculus, if needed, to calculate each of the derivatives given below.
$\frac{d^{2}}{d x^{2}} {鈭�}_{x^{2}}^{1} 鈥� \ln 鈦� | t | d t$