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

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Chapter 1: Problem 7

Find \(d^{2} y / d x^{2}\). $$y=7 x+2$$

Short Answer

Expert verified
The second derivative is zero: $$\frac{d^2y}{dx^2} = 0$$.

Step by step solution

01

Identify the Function

The given function is a linear equation: $$y = 7x + 2$$
02

- Find the First Derivative

To find the second derivative, start by finding the first derivative of the given function. The derivative of a linear function $$y = ax + b$$ is simply the coefficient of $$x$$. For $$y = 7x + 2$$, $$\frac{dy}{dx} = 7$$.
03

- Find the Second Derivative

Since the first derivative is a constant, the derivative of a constant term will be zero. $$\frac{d^2y}{dx^2} = 0$$.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

calculus
Calculus is a branch of mathematics that focuses on studying change. It's divided into differential calculus and integral calculus. Differential calculus deals with the concept of the derivative, which is the rate at which a function changes at any given point.

The given problem asks us to find the second derivative of the function. This involves calculating how the rate of change itself changes. In simpler terms, we are looking at the acceleration when given a position. For our specific function, this will involve a few straightforward steps of differentiation.
linear functions
A linear function is one of the simplest types of functions in mathematics. It has the general form:

$$ y = mx + b $$
Where:
  • $$ m $$ is the slope or rate of change.
  • $$ x $$ is the independent variable.
  • $$ b $$ is the y-intercept or the point where the line crosses the y-axis.


In our problem, the linear function provided is $$ y = 7x + 2 $$. Here, the slope $$ m $$ is 7, which indicates that for every unit increase in $$ x $$, $$ y $$ increases by 7. The y-intercept $$ b $$ is 2. This means the line crosses the y-axis at 2. Understanding these components helps in differentiating the function.
differentiation
Differentiation is the process of finding the derivative of a function. The derivative represents the rate of change of a function with respect to a variable. For a function $$ f(x) $$, the first derivative is denoted as $$ f'(x) $$ or $$ \frac{dy}{dx} $$. To find it, follow these steps:

  1. Identify the function to differentiate. In our case, $$ y = 7x + 2 $$.
  2. Apply the differentiation rules. For linear functions of the form $$ y = mx + b $$, the derivative is simply the slope $$ m $$. So, $$ \frac{dy}{dx} = 7 $$.
  3. To find the second derivative, differentiate the first derivative. Since the first derivative of 7 is a constant, its derivative is 0. Thus, $$ \frac{d^2y}{dx^2} = 0 $$.


You can see that the second derivative of a linear function is always zero because the rate of change of the slope (which is constant) does not change.

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Most popular questions from this chapter

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