Chapter 2: Q. 47 (page 233)

Find the derivatives of each of the functions in Exercises 17–50. In some cases it may be convenient to do some preliminary algebra
$f (x) = \ln (arcsec (\sin^{2} x))$

Short Answer

Expert verified

The derivative is $\frac{2 \sin x \cos x}{arcsec (\sin^{2} x)} (\frac{1}{\sin^{2} x \sqrt{\sin^{4} x - 1}})$ .

Step by step solution

Step 1. Given Information.

The given function is,

$f (x) = \ln (arcsec (\sin^{2} x))$ .

Step 2. Preliminary Algebra.

$C h a i n R u l e : (f \circ u)^{'} (x) = f^{'} (u (x)) u^{'} (x) \frac{d}{d x} (\sec^{- 1} x) = \frac{1}{| x | \sqrt{x^{2} - 1}} \frac{d}{d x} (\sin x) = \cos x$ We know,

Step 3. Derivative of the function.

The derivative of the function is,

$\frac{d}{d x} \ln (arcsec (\sin^{2} x)) = \frac{1}{arcsec (\sin^{2} x)} \frac{d}{d x} (arcsec (\sin^{2} x)) = \frac{1}{arcsec (\sin^{2} x)} (\frac{1}{|\sin^{2} x| \sqrt{{(\sin^{2} x)}^{2} - 1}}) \frac{d}{d x} (\sin^{2} x) = \frac{1}{arcsec (\sin^{2} x)} (\frac{1}{|\sin^{2} x| \sqrt{\sin^{4} x - 1}}) (2 \sin x) \frac{d}{d x} (\sin x) = \frac{1}{arcsec (\sin^{2} x)} (\frac{1}{\sin^{2} x \sqrt{\sin^{4} x - 1}}) (2 \sin x \cos x)$

$= \frac{2 \sin x \cos x}{arcsec (\sin^{2} x)} (\frac{1}{\sin^{2} x \sqrt{\sin^{4} x - 1}})$

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.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

Recommended explanations on Math Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Find the derivatives of each of the functions in Exercises 17–50. In some cases it may be convenient to do some preliminary algebra
$f (x) = \ln (arcsec (\sin^{2} x))$

Short Answer

Step by step solution

Step 1. Given Information.

Step 2. Preliminary Algebra.

Step 3. Derivative of the function.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Pure Maths

Geometry

Decision Maths

Calculus

Discrete Mathematics

Theoretical and Mathematical Physics

Study anywhere. Anytime. Across all devices.

Find the derivatives of each of the functions in Exercises 17–50. In some cases it may be convenient to do some preliminary algebraf(x)=lnarcsecsin2x

Short Answer

Step by step solution

Step 1. Given Information.

Step 2. Preliminary Algebra.

Step 3. Derivative of the function.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Pure Maths

Geometry

Decision Maths

Calculus

Discrete Mathematics

Theoretical and Mathematical Physics

Study anywhere. Anytime. Across all devices.

Find the derivatives of each of the functions in Exercises 17–50. In some cases it may be convenient to do some preliminary algebra
$f (x) = \ln (arcsec (\sin^{2} x))$