/*! 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 3 Find \(d^{2} y / d x^{2}\). $$... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 1: Problem 3

Find \(d^{2} y / d x^{2}\). $$y=2 x^{4}-5 x$$

Short Answer

Expert verified
\( \frac{d^2y}{dx^2} = 24x^2 \)

Step by step solution

01

Identify the function

The given function is \[ y = 2x^4 - 5x \]. The goal is to find the second derivative of this function.
02

Find the first derivative

To find the first derivative, differentiate the function with respect to x. Use the power rule \( \frac{d}{dx} [x^n] = nx^{n-1} \). The first derivative of \[ y = 2x^4 - 5x \] is calculated as follows: \[ \frac{dy}{dx} = \frac{d}{dx} (2x^4) - \frac{d}{dx} (5x) = 8x^3 - 5 \].
03

Find the second derivative

Differentiate the first derivative \( \frac{dy}{dx} = 8x^3 - 5 \) with respect to x to obtain the second derivative, again applying the power rule: \[ \frac{d^2y}{dx^2} = \frac{d}{dx} (8x^3) - \frac{d}{dx} (5) = 24x^2 \].

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 Differentiation
Understanding how to apply the power rule in differentiation is fundamental in calculus. The power rule states that if you have a function of the form \[ f(x) = x^n \], where \[ n \] is a real number, then its derivative is \[ f'(x) = nx^{n-1} \]. This means you multiply the exponent by the coefficient from the original term and then subtract one from the exponent.
For example:
  • If \[ f(x) = x^3 \], applying the power rule gives us \[ f'(x) = 3x^2 \].
  • If \[ f(x) = 2x^4 \], the derivative would be \[ f'(x) = 8x^3 \].
Notice how applying this rule is straightforward. Just remember: multiply first, then reduce the exponent by one.
Finding the First Derivative
Finding the first derivative of a function is the initial step in solving many calculus problems. The first derivative represents the rate of change of the function with respect to its variable, often providing information about the slope of the function.
The given function is \[ y = 2x^4 - 5x \]. To find its first derivative, we'll apply the power rule:
  • Differentiate \[ 2x^4 \]: \[ 4 \times 2 \times x^{4-1} = 8 \times x^3 = 8x^3 \]
  • Differentiate \[ -5x \]: \[ -5 \times 1 \times x^{1-1} = -5 \]

    • Combining these, the first derivative is:
    \[ \frac{dy}{dx} = 8x^3 - 5 \].
    This provides the slope of the function at any point \[ x \].
Finding the Second Derivative
The second derivative provides information on the curvature or concavity of the function. It tells us how the rate of change of the first derivative is changing. This helps in understanding the behavior of the function in more detail.
To find the second derivative, we differentiate the first derivative we just found:
Recall that:
  • First derivative: \[ \frac{dy}{dx} = 8x^3 - 5 \]
Now, apply the power rule again:
  • Differentiate \[ 8x^3 \]: \[ 3 \times 8 \times x^{3-1} = 24 \times x^2 = 24x^2 \]
  • Differentiate the constant \[ -5 \]: Any constant's derivative is 0

Thus, the second derivative is:
\[ \frac{d^2y}{dx^2} = 24x^2 \].
This shows how the slope of the original function changes as \[ x \] varies, giving a deeper insight into the function's shape.

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

Tongue-Tied Sauces, Inc., finds that the revenue, in dollars, from the sale of \(x\) bottles of barbecue sauce is given by \(R(x)=7.5 x^{0.7} .\) Find the rate at which average revenue is changing when 81 bottles of barbecue sauce have been produced.

The temperature T of a person during an illness is given by \(T(t)=\frac{4 t}{t^{2}+1}+98.6\) where \(T\) is the temperature, in degrees Fahrenheit, at time \(t,\) in hours. (GRAPH CANNOT COPY)

Differentiate each function. $$g(t)=\frac{-t^{2}+3 t+5}{t^{2}-2 t+4}$$

For each function, find the interval(s) for which \(f^{\prime}(x)\) is positive. Find the points on the graph of \(y=x^{4}-\frac{4}{3} x^{2}-4\) at which the tangent line is horizontal.

Differentiate each function. $$f(x)=\frac{3 x^{2}+2 x}{x^{2}+1}$$
See all solutions

Recommended explanations on Math Textbooks

Theoretical and Mathematical Physics

Read Explanation

Pure Maths

Read Explanation

Discrete Mathematics

Read Explanation

Applied Mathematics

Read Explanation

Logic and Functions

Read Explanation

Mechanics Maths

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.