Chapter 2: Q. 33 (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.
$f (x) = \frac{3 x^{2} \ln x}{\tan x} .$

Short Answer

Expert verified

The derivative of the given function is: $\frac{3 x \tan x (1 + 2 \ln x) - 3 x^{2} \ln x s e c^{2} x}{\tan^{2} x} .$

Step by step solution

Step 1. Given Information.

$f (x) = \frac{3 x^{2} \ln x}{\tan x} .$

Step 2. General formulas for finding derivatives.

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

Step 3. Solving derivative using above formulas.

$f (x) = \frac{3 x^{2} \ln x}{\tan x}, D i f f e r e n t i a t i n g u \sin g p r o d u c t a n d q u o t i e n t r u l e w e g e t, \frac{d}{d x} (f (x)) = \frac{\tan x \frac{d}{d x} (3 x^{2} \ln x) - (3 x^{2} \ln x) \frac{d}{d x} (\tan x)}{\tan^{2} x} = \frac{\tan x (3 x^{2} \frac{1}{x} + 6 x \ln x) - (3 x^{2} \ln x) (s e c^{2} x)}{\tan^{2} x} = \frac{\tan x (3 x + 6 x \ln x) - (3 x^{2} \ln x) s e c^{2} x}{\tan^{2} x} = \frac{3 x \tan x (1 + 2 \ln x) - 3 x^{2} \ln x s e c^{2} x}{\tan^{2} x} .$

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Find the derivatives of each of the functions in Exercises 17–50. In some cases it may be convenient to do some preliminary algebra.
$f (x) = \frac{3 x^{2} \ln x}{\tan x} .$

Short Answer

Step by step solution

Step 1. Given Information.

Step 2. General formulas for finding derivatives.

Step 3. Solving derivative using above formulas.

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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)=3x2lnxtanx.

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

Statistics

Discrete Mathematics

Decision Maths

Pure Maths

Probability and Statistics

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.
$f (x) = \frac{3 x^{2} \ln x}{\tan x} .$