/*! 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 8 Find \(d^{2} y / d x^{2}\). $$... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 1: Problem 8

Find \(d^{2} y / d x^{2}\). $$y=6 x-3$$

Short Answer

Expert verified
The second derivative is 0.

Step by step solution

01

Understand the Problem

The task is to find the second derivative of the function. First, identify the given function: \[ y = 6x - 3 \]
02

Compute the First Derivative

To find the second derivative, start by computing the first derivative using basic differentiation rules. Differentiating \( y \) with respect to \( x \): \[ \frac{dy}{dx} = \frac{d}{dx}(6x - 3) = 6 \]
03

Compute the Second Derivative

Now, obtain the second derivative by differentiating the first derivative: \[ \frac{d^2 y}{dx^2} = \frac{d}{dx}(6) = 0 \]

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.

Differentiation
Differentiation is a fundamental concept in calculus. It involves finding the derivative of a function, which represents the rate of change of the function's value with respect to a variable, often denoted as \(x\). Differentiation is crucial in solving many real-world problems such as calculating speed, acceleration, and optimizing functions. To differentiate a function, we apply specific rules based on the type of function. The process is systematic and often begins with identifying the function to differentiate.
First Derivative
To find the first derivative of a function means to compute the slope of the tangent line to the function at any given point. This gives insight into how the function's value changes as the independent variable changes. For example, given \(y = 6x - 3\), the first derivative is calculated by applying the power rule and constant rule. We differentiate \(6x\) to get 6 (since the derivative of \(x\) with respect to \(x\) is 1, and 6 is a constant multiplier), and the derivative of the constant term \(-3\) is 0. Therefore, \( \frac{dy}{dx} = 6 \). This reveals that the rate of change of \(y\) with respect to \(x\) is consistently 6.
Basic Differentiation Rules
Basic differentiation rules are essential tools for finding derivatives efficiently. Some of the fundamental rules include:
  • The Power Rule: If \(y = x^n\), then \( \frac{dy}{dx} = nx^{n-1} \).
  • The Constant Rule: The derivative of a constant is zero.
  • The Constant Multiple Rule: If \(y = c \times f(x)\), then \( \frac{dy}{dx} = c \times f'(x) \), where \(c\) is a constant.

In the given exercise, we used the Constant Multiple Rule and the Constant Rule to find the first and second derivatives. First, we differentiated \(6x - 3\) to get 6, then we took the derivative of 6, a constant, to get 0. These rules simplify the process and are foundational to more advanced differentiation techniques.

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 \(d y / d x .\) Each function can be differentiated using the rules developed in this section, but some algebra may be required beforehand. $$y=(\sqrt{x}+\sqrt[3]{x})^{2}$$

The following is the beginning of an alternative proof of the Quotient Rule that uses the Product Rule and the Power Rule. Complete the proof, giving reasons for each step. Proof. Let $$Q(x)=\frac{N(x)}{D(x)}$$ Then $$Q(x)=N(x) \cdot[D(x)]^{-1}$$ Therefore,

The function \(f(x)=x^{3}+a x\) is always increasing if \(a>0,\) but not if \(a<0 .\) Use the derivative of \(f\) to explain why this observation is true.

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. $$y=\left(\frac{x}{\sqrt{x-1}}\right)^{3}$$
See all solutions

Recommended explanations on Math Textbooks

Calculus

Read Explanation

Pure Maths

Read Explanation

Geometry

Read Explanation

Logic and Functions

Read Explanation

Applied Mathematics

Read Explanation

Statistics

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.