Q. 47 Find the derivatives of each of ... [FREE SOLUTION]

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Chapter 2: Q. 47 (page 233)

Find the derivatives of each of the functions in Exercises 17鈥�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.

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

Step 3. Derivative of the function.

The derivative of the function is,

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

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

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91影视

Find the derivatives of each of the functions in Exercises 17鈥�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.

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91影视

Find the derivatives of each of the functions in Exercises 17鈥�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

Calculus

Geometry

Theoretical and Mathematical Physics

Logic and Functions

Probability and Statistics

Decision Maths

Study anywhere. Anytime. Across all devices.

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