/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 8 Explain the general procedure of... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 3: Problem 8

Explain the general procedure of logarithmic differentiation.

Short Answer

Expert verified
Answer: The main steps in logarithmic differentiation are: 1. Start with the given function y = f(x). 2. Take the natural logarithm (ln) of both sides: ln(y) = ln(f(x)). 3. Apply logarithmic properties to simplify the equation. 4. Differentiate both sides with respect to x using the chain rule. 5. Solve for dy/dx to find the derivative of the original function. 6. Express the derivative in terms of the original function y = f(x).

Step by step solution

01

Start with the function

Given a function y = f(x), which may have variables in the exponents or be a function of another function, that we want to differentiate.
02

Take the natural logarithm of both sides

In order to apply logarithmic differentiation, we need to take the natural logarithm (ln) of both sides of the equation to get: ln(y) = ln(f(x))
03

Apply logarithmic properties

Use the properties of logarithms to simplify the equation as much as possible. Some useful properties include: 1. ln(a * b) = ln(a) + ln(b) 2. ln(a / b) = ln(a) - ln(b) 3. ln(a^b) = b * ln(a)
04

Differentiate both sides

Now that the equation has been simplified using logarithmic properties, differentiate both sides with respect to x. On the left side, use the chain rule for differentiation: d[ln(y)]/dx = d[ln(y)]/dy * dy/dx On the right side, differentiate the simplified f(x) term-by-term.
05

Solve for dy/dx

As a result of differentiation, you now have an equation involving dy/dx. Isolate dy/dx in the equation to find the derivative of the original function, y = f(x).
06

Express the derivative in terms of the original function

If required, express the derivative dy/dx in terms of the original function y = f(x). This may involve rewriting the derivative to make it look more familiar, or replacing the original y term with f(x).

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

Most popular questions from this chapter

Product Rule for the second derivative Assuming the first and second derivatives of \(f\) and \(g\) exist at \(x\), find a formula for \(\frac{d^{2}}{d x^{2}}(f(x) g(x))\)

Consider the following functions (on the given interval, if specified). Find the inverse function, express it as a function of \(x,\) and find the derivative of the inverse function. $$f(x)=x^{2 / 3}, \text { for } x>0$$

The bottom of a large theater screen is \(3 \mathrm{ft}\) above your eye level and the top of the screen is \(10 \mathrm{ft}\) above your eye level. Assume you walk away from the screen (perpendicular to the screen) at a rate of \(3 \mathrm{ft} / \mathrm{s}\) while looking at the screen. What is the rate of change of the viewing angle \(\theta\) when you are \(30 \mathrm{ft}\) from the wall on which the screen hangs, assuming the floor is horizontal (see figure)?

Visualizing tangent and normal lines a. Determine an equation of the tangent line and normal line at the given point \(\left(x_{0}, y_{0}\right)\) on the following curves. (See instructions for Exercises \(63-68 .)\) b. Graph the tangent and normal lines on the given graph. \(\left(x^{2}+y^{2}-2 x\right)^{2}=2\left(x^{2}+y^{2}\right);\) \(\left(x_{0}, y_{0}\right)=(2,2)\) (lima莽on of Pascal)

Identifying functions from an equation The following equations implicitly define one or more functions. a. Find \(\frac{d y}{d x}\) using implicit differentiation. b. Solve the given equation for \(y\) to identify the implicitly defined functions \(y=f_{1}(x), y=f_{2}(x), \ldots\) c. Use the functions found in part (b) to graph the given equation. \(x+y^{3}-x y=1\) (Hint: Rewrite as \(y^{3}-1=x y-x\) and then factor both sides.)
See all solutions

Recommended explanations on Math Textbooks

Mechanics Maths

Read Explanation

Pure Maths

Read Explanation

Logic and Functions

Read Explanation

Discrete Mathematics

Read Explanation

Applied Mathematics

Read Explanation

Theoretical and Mathematical Physics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.