/*! 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 7 In Exercises 1 through \(34,\) f... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 4: Problem 7

In Exercises 1 through \(34,\) find the derivative. $$ f(x)=\sqrt[3]{x^{2}} $$

Short Answer

Expert verified
The derivative is \( f'(x) = \frac{2}{3\sqrt[3]{x}} \).

Step by step solution

01

Rewrite the function

Start by rewriting the cubic root in a form that is easily differentiable. For the function \( f(x) = \sqrt[3]{x^2} \), change it to exponential form: \( f(x) = x^{2/3} \). This helps us apply the power rule later.
02

Apply the Power Rule

Use the power rule for derivatives, which states that \( \frac{d}{dx}(x^n) = nx^{n-1} \). Here, \( n = \frac{2}{3} \), so applying the power rule gives: \( f'(x) = \frac{2}{3}x^{\frac{2}{3} - 1} \).
03

Simplify the derivative

Simplify the expression by calculating the exponent: \( \frac{2}{3} - 1 = -\frac{1}{3} \). Therefore, the derivative is \( f'(x) = \frac{2}{3}x^{-\frac{1}{3}} \).
04

Rewrite the derivative in radical form

Often, it's preferable to write the answer in the original form as a radical. We convert \( x^{-1/3} \) back to a radical: \( x^{-1/3} = \frac{1}{\sqrt[3]{x}} \). Therefore, the derivative is: \( f'(x) = \frac{2}{3\sqrt[3]{x}} \).

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.

Power Rule
The power rule is a fundamental rule in calculus used to find the derivative of functions of the form \(x^n\). Derivatives represent the rate at which a function changes. The power rule simplifies the process of differentiation by providing a straightforward formula:
  • To differentiate \(x^n\), multiply the exponent by the coefficient, \(n\), then decrease the exponent by one.
  • The power rule formula is: \(\frac{d}{dx}(x^n) = nx^{n-1}\).
  • This means if \(f(x) = x^{2/3}\), then the derivative \(f'(x)\) would be \(\frac{2}{3}x^{-1/3}\).
Understanding this rule allows you to quickly find derivatives of polynomial functions without extensive algebraic manipulations. It鈥檚 particularly helpful when dealing with functions that can be rewritten in exponential form.
Exponential Form
Converting expressions into exponential form makes differentiation much easier. This is especially true for expressions involving roots, such as finding derivatives of functions with variables inside a radical. By using exponential notation, calculations become straightforward with the application of the power rule.
  • For example, the cubic root \(\sqrt[3]{x^2}\) is converted to \(x^{2/3}\). This new form is the exponential expression.
  • This step is crucial because it transforms a potentially complex root form into a simpler exponential format, aligning it perfectly for differentiation with the power rule.
  • Remember that the process involves expressing roots as fractional exponents.
Being comfortable with changing between radical and exponential forms enhances your ability to tackle various calculus problems effectively.
Radical Form
Returning to radical form after differentiating can often make the result more recognizable and easier to interpret. This is especially helpful in presenting your final answer in a form that matches the original function's style.
  • In our example, after finding the derivative as \(\frac{2}{3}x^{-1/3}\), you can rewrite it as \(\frac{2}{3\sqrt[3]{x}}\).
  • To convert back, remember that a negative exponent indicates division: \(x^{-1/3}\) is the same as \(\frac{1}{x^{1/3}}\).
  • Thus \(x^{1/3}\) corresponds to \(\sqrt[3]{x}\), the cubic root.
This conversion step is beneficial because it makes the mathematical result more accessible and easier to communicate, especially in a teaching or learning context.

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

In Exercises 35 through \(38,\) find \(d y / d x\). $$ y=\sqrt{x}+\frac{1}{\sqrt{x}} $$

Biology Pilarska \(^{33}\) used the equation $$ y=15.97 \frac{x}{1.47+x} $$ to model the ingestion rate \(y\) of an individual female rotifer as a function of the density \(x\) of green algae it feeds on. Determine the rate of change of the ingestion rate with respect to the density. What is the sign of this rate of change. What is the significance of this sign?

Biology The chemical d-limonene is extremely toxic to some insects. Karr and Coats \(^{22}\) showed that the number of days \(y\) for a German cockroach nymph (Blatta germanica) to reach the adult stage when given a diet containing this chemical was approximated by \(y=f(x)=\) \(125-3.43 \ln x,\) where \(x\) is the percent of this chemical in the diet. Find \(f^{\prime}(x)\) for any \(x>0\). Interpret what this means. What does this say about the effectiveness of this chemical as a control of the German cockroach nymph? Graph \(y=f(x)\)

Lactation Curves Cardellino and Benson \(^{10}\) studied lactation curves for ewes rearing lambs. They created the mathematical model given by the equation \(P(D)=273.23+\) \(5.39 D-0.1277 D^{2},\) where \(D\) is day of lactation and \(P\) is milk production in grams. Find \(P^{\prime}(D)\). Find \(P^{\prime}(30)\). Give units.

Biology Zonneveld and Kooijman \(^{17}\) determined that the equation \(y=2.81 L^{2}\) approximated the lettuce intake versus the shell length of a pond snail. Find the rate of change of lettuce intake with respect to shell length.
See all solutions

Recommended explanations on Math Textbooks

Applied Mathematics

Read Explanation

Mechanics Maths

Read Explanation

Logic and Functions

Read Explanation

Probability and Statistics

Read Explanation

Statistics

Read Explanation

Geometry

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.