/*! 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 16 Differentiate the functions. $... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 3: Problem 16

Differentiate the functions. $$y=x^{4}+4 \sqrt[4]{x}$$

Short Answer

Expert verified
\(\frac{dy}{dx} = 4x^3 + x^{-3/4}\)

Step by step solution

01

Identify given function components

The function to differentiate is given as \(y = x^4 + 4\root{4}{x}\). It has two components: \(x^4\) and \(4\root{4}{x}\).
02

Rewrite fractional exponent

Rewrite the second term with a fractional exponent: \(y = x^4 + 4x^{1/4}\).
03

Differentiate each term separately

Differentiate each term using the power rule, \(\frac{d}{dx} x^n = nx^{n-1}\). For \(x^4\), the derivative is \(4x^3\). For \(4x^{1/4}\), the derivative is \(4 \cdot \frac{1}{4} x^{1/4-1} = x^{-3/4}\).
04

Combine the derivatives

Add the results from each differentiation step to get the final derivative: \(\frac{dy}{dx} = 4x^3 + x^{-3/4}\).

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 Explained
The power rule is a simple but crucial concept in calculus for differentiation. To apply the power rule, you use the formula: \[ \frac{d}{dx} x^n = nx^{n-1} \]. This means you multiply the power (n) by the term, and then subtract one from the power to find the derivative.

For example, if you need to differentiate \( x^4 \), you multiply 4 (the exponent) by \( x \), and then reduce the exponent by 1: \[ \frac{d}{dx} x^4 = 4x^{4-1} = 4x^3 \].

This rule helps to simplify differentiation, making it more straightforward to handle polynomial terms. In our exercise, we took the derivative of \( x^4 \) easily with this method.
Understanding Fractional Exponents
Fractional exponents can seem tricky at first, but they follow the same differentiation rules as whole number exponents.

In our function, we dealt with \( 4\root{4}{x} \). First, we rewrote it using a fractional exponent: \[ 4\root{4}{x} = 4x^{1/4} \].

To differentiate \( 4x^{1/4} \), we use the power rule: \[ \frac{d}{dx} 4x^{1/4} = 4 \times \frac{1}{4} x^{1/4 - 1} = x^{-3/4} \].

By rewriting roots as fractional exponents, you can simplify many differentiation problems, making them easier to solve.

This is especially helpful for more complex functions.
Combining Derivatives
After differentiating each term separately, the next step is combining the derivatives to find the overall derivative of the function.

For the given function, the derivatives of the individual terms \( x^4 \) and \( 4x^{1/4} \) are \( 4x^3 \) and \( x^{-3/4} \), respectively.

Combining these, we get: \[ \frac{dy}{dx} = 4x^3 + x^{-3/4} \].

This step involves simply adding the differentiated terms together.

Understanding how to combine derivatives helps you to handle more sophisticated functions that have multiple components. This approach maintains the structure of the original function while applying the principles of differentiation.

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

Find the equation of the tangent line to the graph of \(x^{4} y^{2}=144\) at the point \((2,3)\) and at the point \((2,-3).\)

The monthly advertising revenue, \(A,\) and the monthly circulation, \(x,\) of a magazine are related approximately by the equation $$A=6 \sqrt{x^{2}-400}, \quad x \geq 20,$$ where \(A\) is given in thousands of dollars and \(x\) is measured in thousands of copies sold. At what rate is the advertising revenue changing if the current circulation is \(x=25\) thousand copies and the circulation is growing at the rate of 2 thousand copies per month? \(\left[\text {Hint: Use the chain rule } \frac{d A}{d t}=\frac{d A}{d x} \frac{d x}{d t}\right].\)

Suppose that in Boston the wholesale price, \(p,\) of oranges (in dollars per crate) and the daily supply, \(x\) (in thousands of crates), are related by the equation \(p x+7 x+8 p=328 .\) If there are 4 thousand crates available today at a price of \(\$ 25\) per crate, and if the supply is changing at the rate of -.3 thousand crates per day, at what rate is the price changing?

Use implicit differentiation of the equations to determine the slope of the graph at the given point. $$\sqrt{x}+\sqrt{y}=7 ; x=9, y=16$$

Use implicit differentiation of the equations to determine the slope of the graph at the given point. $$y^{2}=x^{3}+1 ; x=2, y=-3$$
See all solutions

Recommended explanations on Math Textbooks

Decision Maths

Read Explanation

Mechanics Maths

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Pure Maths

Read Explanation

Applied Mathematics

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.