Q. 32 Find the derivatives of each of ... [FREE SOLUTION]

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Chapter 2: Q. 32 (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.
$f (x) = \frac{\sin x c s c x}{c o t x \cos x} .$

Short Answer

Expert verified

The derivative of the given function is: $\frac{1 + \sin^{2} x}{\cos^{3} x} .$

Step by step solution

Step 1. Given Information.

$f (x) = \frac{\sin x c s c x}{c o t x \cos x} .$

Step 2. General formulas for finding derivatives.

$\frac{d}{d x} (x^{n}) = n x^{n - 1}, \frac{d}{d x} (\sin x) = \cos x, \frac{d}{d x} (\cos x) = - \sin x, \frac{d}{d x} (\tan x) = s e c^{2} x, \frac{d}{d x} (s e c x) = s e c x \tan x, \frac{d}{d x} (c s c x) = - c s c x c o t x, \frac{d}{d x} (c o t x) = - c s c^{2} x, P r o d u c t r u l e : \frac{d}{d x} (u v) = u \frac{d v}{d x} + v \frac{d u}{d x}, Q u o t i e n t r u l e : \frac{d}{d x} (\frac{u}{v}) = \frac{v \frac{d u}{d x} - u \frac{d v}{d x}}{v^{2}}, C h a i n r u l e : \frac{d}{d x} (f o g (x)) = \frac{d}{d x} (f (g (x))) 脳 \frac{d}{d x} (g (x) .$

Step 3. Solving derivative using above formulas.

$f (x) = \frac{\sin x c s c x}{c o t x \cos x}, S o l v i n g t h e f u n c t i o n f i r s t, f (x) = \frac{\tan x}{\cos x}, N o w d i f f e r e n t i a t i n g u \sin g q u o t i e n t r u l e w e g e t, \frac{d}{d x} (f (x)) = \frac{\cos x \frac{d}{d x} (\tan x) - \tan x \frac{d}{d x} (\cos x)}{\cos^{2} x} = \frac{\cos x (s e c^{2} x) - \tan x (- \sin x)}{\cos^{2} x} = \frac{s e c x + \tan x \sin x}{\cos^{2} x} = \frac{1 + \sin^{2} x}{\cos^{3} x} .$

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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.
$f (x) = \frac{\sin x c s c x}{c o t x \cos x} .$

Short Answer

Step by step solution

Step 1. Given Information.

Step 2. General formulas for finding derivatives.

Step 3. Solving derivative using above formulas.

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 algebra.f(x)=sinxcscxcotxcosx.

Short Answer

Step by step solution

Step 1. Given Information.

Step 2. General formulas for finding derivatives.

Step 3. Solving derivative using above formulas.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Geometry

Applied Mathematics

Theoretical and Mathematical Physics

Calculus

Decision Maths

Pure 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.
$f (x) = \frac{\sin x c s c x}{c o t x \cos x} .$