/*! 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 45 $$ \text { Differentiate. } ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 4: Problem 45

$$ \text { Differentiate. } $$ $$ y=x^{4} \ln x-\frac{1}{2} x^{2} $$

Short Answer

Expert verified
The derivative is \( y' = 4x^3 \ln x + x^3 - x \).

Step by step solution

01

- Understand Differentiation Rules

To differentiate the given function, recognize that it is composed of different terms: the product of two functions and a simple power function. You will need to apply the product rule and the power rule.
02

- Apply the Product Rule

The product rule states that to differentiate the product of two functions, say \(f(x)g(x)\), you use the formula \((f \, g)' = f' \, g + f \, g'\). Here, let \(f(x) = x^4\) and \(g(x) = \ln x\). Find the derivatives of these functions separately.\[ f'(x) = 4x^3 \]\[ g'(x) = \frac{1}{x} \] Applying the product rule: \[ \frac{d}{dx} [x^4 \, \ln x] = 4x^3 \ln x + x^4 \cdot \frac{1}{x} = 4x^3 \ln x + x^3 \]
03

- Differentiate the Power Rule Term

Next, apply the power rule to the second term, which is \(-\frac{1}{2} x^2\). The power rule states that for any term \(x^n\), the derivative is \(nx^{n-1}\). Applying this rule, \[ \frac{d}{dx} \left( -\frac{1}{2} x^2 \right) = -\frac{1}{2} \cdot 2x = -x \]
04

- Combine the Results

Combine the derivatives found in both steps. Thus, the derivative of the given function is: \[ y' = 4x^3 \ln x + x^3 - x \]

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!

Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding Derivatives and Differentiation
To understand the concept of differentiation, let's first define what a **derivative** is. The derivative of a function measures how the function's output value changes as its input value changes. In simpler terms, it tells us the rate of change or slope of the function at any given point. Differentiation is the process of finding the derivative.

For instance, if you have a function like \( y = x^4 \ln x - \frac{1}{2} x^2 \), finding its derivative involves calculating how each term of the function changes concerning \( x \).
Applying the Product Rule
The **product rule** is used in differentiation when you need to find the derivative of the product of two functions. The rule is represented by the formula: \( (f \, g)' = f' \, g + f \, g' \).

So, if you have two functions, say \( f(x) = x^4 \) and \( g(x) = \ln x \), and you want to find the derivative of their product \( x^4 \ln x \), you'd first differentiate each term separately. Here鈥檚 how:
  • The derivative of \( f(x) = x^4 \) is \( f'(x) = 4x^3 \).
  • The derivative of \( g(x) = \ln x \) is \( g'(x) = \frac{1}{x} \).
Following the product rule, we get:

\[ \frac{d}{dx} [x^4 \, \ln x] = 4x^3 \ln x + x^4 \cdot \frac{1}{x} = 4x^3 \ln x + x^3 \]
Using the Power Rule
The **power rule** is a basic rule in differentiation that makes it easier to find the derivative of a power function. The rule states that for any term \( x^n \), its derivative is \( nx^{n-1} \).

For example, to differentiate \( -\frac{1}{2} x^2 \), apply the power rule:
  • First, bring down the exponent: \( -\frac{1}{2} \cdot 2 \).
  • Then, subtract one from the exponent: \( x^{2-1} = x \).
Applying these steps, we get:

\[ \frac{d}{dx} \left( -\frac{1}{2} x^2 \right) = -x \]

By combining the results of the product rule and power rule, you get the overall derivative:

\[ y' = 4x^3 \ln x + x^3 - x \]

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

A student made the following error on a test: $$ \frac{d}{d x} e^{x}=x e^{x-1} $$ Explain the error and how to correct it.

The coroner arrives at the scene of a murder at 2 A.M. He takes the temperature of the body and finds it to be \(61.6^{\circ}\). He waits \(1 \mathrm{hr}\), takes the temperature again, and finds it to be \(57.2^{\circ}\). The body is in a meat freezer, where the temperature is \(10^{\circ}\). When was the murder committed?

complete the following. Central America

Differentiate. $$ f(x)=x(5.4)^{x} $$

A bank advertises that it compounds interest continuously and that it will double your money in 10 yr. What is its exponential growth rate?
See all solutions

Recommended explanations on Biology Textbooks

Genetic Information

Read Explanation

Plant Biology

Read Explanation

Biology Experiments

Read Explanation

Ecological Levels

Read Explanation

Communicable Diseases

Read Explanation

Biological Molecules

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.