Q. 64 Use logarithmic differentiation ... [FREE SOLUTION]

91影视

Chapter 2: Q. 64 (page 234)

Use logarithmic differentiation to find derivative
$f (x) = (\sec 鈦� x)^{x}$

Short Answer

Expert verified

The answer of this question is

$f^{鈥�} (x) = (\ln 鈦� (\sec 鈦� x) + x \tan 鈦� x) (\sec 鈦� x)^{x}$

Step by step solution

Step 1.Given information

We have been given to find the derivative of function

$f (x) = (\sec 鈦� x)^{x}$

Step 2.Derivative of given function

Product rule

$(f g)^{鈥�} (x) = f^{鈥�} (x) g (x) + f (x) g^{鈥�} (x)$

Quotient rule

${(\frac{f}{g})}^{鈥�} (x) = \frac{f^{鈥�} (x) g (x) 鈭� f (x) g^{鈥�} (x)}{(g (x))^{2}}$

Step 3.Using logarithmic approach

Taking log both side and then approaching

$\begin{aligned} f (x) 鈥呪赌呪赌呪赌� = (\sec 鈦� x)^{x} \\ \ln 鈦� f (x) 鈥呪赌呪赌呪赌� = \ln 鈦� (\sec 鈦� x)^{x} [Taking log in both sides] \\ \ln 鈦� f (x) 鈥呪赌呪赌呪赌� = x \ln 鈦� (\sec 鈦� x) \\ \frac{f^{鈥�} (x)}{f (x)} 鈥呪赌呪赌呪赌� = \frac{d}{d x} (x \ln 鈦� (\sec 鈦� x)) [\begin{matrix} Differentiate both sides \\ with respect to x \end{matrix}] \\ \frac{f^{鈥�} (x)}{f (x)} 鈥呪赌呪赌呪赌� = (\frac{d}{d x} (x)) \ln 鈦� (\sec 鈦� x) + x (\frac{d}{d x} \ln 鈦� (\sec 鈦� x)) \\ 鈥呪赌呪赌呪赌� = \ln 鈦� (\sec 鈦� x) + \frac{x}{\sec 鈦� x} (\frac{d}{d x} \sec 鈦� x) [\begin{matrix} Using product rule \\ \frac{d}{d x} \ln 鈦� x = \frac{1}{x} \end{matrix}] \\ 鈥呪赌呪赌呪赌� = \ln 鈦� (\sec 鈦� x) + \frac{x}{\sec 鈦� x} (\sec 鈦� x \tan 鈦� x) \\ f^{鈥�} (x) 鈥呪赌呪赌呪赌� = (\frac{d}{d x} \ln 鈦� (\sec 鈦� x = \sec 鈦� x \tan 鈦� x] \end{aligned}]$

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 logarithmic differentiation to find derivative
$f (x) = (\sec 鈦� x)^{x}$

Short Answer

Step by step solution

Step 1.Given information

Step 2.Derivative of given function

Step 3.Using logarithmic approach

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Logic and Functions

Mechanics Maths

Discrete Mathematics

Pure Maths

Probability and Statistics

Geometry

Study anywhere. Anytime. Across all devices.

91影视

Use logarithmic differentiation to find derivativef(x)=(sec鈦�x)x

Short Answer

Step by step solution

Step 1.Given information

Step 2.Derivative of given function

Step 3.Using logarithmic approach

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Logic and Functions

Mechanics Maths

Discrete Mathematics

Pure Maths

Probability and Statistics

Geometry

Study anywhere. Anytime. Across all devices.

Use logarithmic differentiation to find derivative
$f (x) = (\sec 鈦� x)^{x}$