/*! 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 38 For \(y=x^{5},\) find \(d^{4} y ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 1: Problem 38

For \(y=x^{5},\) find \(d^{4} y / d x^{4}\).

Short Answer

Expert verified
The fourth derivative of \( y = x^{5} \) is \( 120x \).

Step by step solution

01

Identify the function

Given the function is \( y = x^{5} \).
02

Differentiate the function for the first time

To find the first derivative, apply the power rule: \( y' = \frac{d}{dx}(x^{5}) = 5x^{4} \).
03

Differentiate the first derivative to find the second derivative

Apply the power rule again: \( y'' = \frac{d}{dx}(5x^{4}) = 20x^{3} \).
04

Differentiate the second derivative to find the third derivative

Continue with the power rule: \( y''' = \frac{d}{dx}(20x^{3}) = 60x^{2} \).
05

Differentiate the third derivative to find the fourth derivative

Apply the power rule one more time: \( y^{(4)} = \frac{d}{dx}(60x^{2}) = 120x \).

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
In calculus, the power rule is a basic technique used to derive the derivative of a power function. A power function is of the form \(f(x) = x^n\) where n is any real number. The power rule states that the derivative of \(x^n\) is given by \(n \cdot x^{(n-1)}\). This means you multiply the power n by the base x raised to one less power. For instance, if you have \(x^5\), its derivative will be \(5 \cdot x^{4}\). Let's see how this applies further in the problem at hand.
first derivative
Taking the first derivative involves applying the power rule to the given function. Here, the function is \(y=x^5\). Using the power rule, we get \(y' = 5x^{4}\). The first derivative tells us the rate at which the original function, \(y\), is changing at any point x. This is a fundamental concept because understanding the first derivative sets the stage for finding higher-order derivatives.
second derivative
The second derivative gives information about the curvature of the function's graph. It tells us how the slope (first derivative) is changing. To find the second derivative, we differentiate the first derivative \(y' = 5x^4\) again using the power rule. This gives us \(y'' = 20x^{3}\). The second derivative \(y''\) shows us the rate at which the slope is changing with respect to x.
third derivative
Continuing the process of differentiation, the third derivative is found by applying the power rule to the second derivative \(y'' = 20x^3\). This results in \(y''' = 60x^2\). The third derivative can indicate the rate of change of the curvature, but it is not often needed in basic calculus problems. For the sake of completeness, however, we perform this step to work our way toward the fourth derivative.

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 \(\frac{d y}{d x}\) \(y=x^{4}-7 x\)

Differentiate each function. $$f(x)=\frac{3 x^{2}-5 x}{x^{2}-1}$$

For each of the following, graph \(f\) and \(f^{\prime}\) and then determine \(f^{\prime}(1) .\) For Exercises use Deriv on the \(T I-83\). $$f(x)=20 x^{3}-3 x^{5}$$

Find the points on the graph of \(y=2 x^{6}-x^{4}-2\) at which the tangent line is horizontal.

If \(f(x)\) is a function, then \((f \circ f)(x)=f(f(x))\) is the composition of \(f\) with itself. This is called an iterated function, and the composition can be repeated many times. For example, \((f \circ f \circ f)(x)=f(f(f(x))) .\) Iterated functions are very useful in many areas, including finance (compound interest is \(a\) simple case) and the sciences (in weather forecasting, for example). For the each function, use the Chain Rule to find the derivative.. If \(f(x)=x^{2}+1,\) find \(\frac{d}{d x}[(f \circ f \circ f)(x)]\).
See all solutions

Recommended explanations on Math Textbooks

Calculus

Read Explanation

Mechanics Maths

Read Explanation

Decision Maths

Read Explanation

Pure Maths

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Logic and Functions

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.