/*! 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 24 Find \(\frac{d y}{d x}\). $$y=... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 1: Problem 24

Find \(\frac{d y}{d x}\). $$y=\frac{3 x}{4}$$

Short Answer

Expert verified
\( \frac{d y}{d x} = \frac{3}{4} \)

Step by step solution

01

Identify the function

The function given is \( y = \frac{3x}{4} \). It is a simple linear function.
02

Differentiate using basic rules

To find the derivative of \( y \) with respect to \( x \), use the constant multiple rule. The constant multiple rule states that if \( y = kx \), where \( k \) is a constant, then \( \frac{d y}{d x} = k \).
03

Apply the constant multiple rule

Here, \( y = \frac{3x}{4} \) can be rewritten as \( y = \frac{3}{4} \times x \). The constant \( k \) is \( \frac{3}{4} \). Hence, the derivative \( \frac{d y}{d x} = \frac{3}{4} \).

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.

constant multiple rule
The constant multiple rule is one of the fundamental rules in differentiation. This rule states that if you have a function of the form \( y = kx \), where \( k \) is a constant, the derivative with respect to \( x \) is simply the constant \( k \).
This can be written as: \[ \frac{d}{dx}(kx) = k \]
This rule simplifies the process of differentiation significantly when dealing with linear functions multiplied by constants.
In our original exercise, \( y = \frac{3x}{4}\), the constant multiple rule makes finding the derivative straightforward. By determining that the constant multiple is \( \frac{3}{4} \), we can readily write the derivative as \( \frac{3}{4} \).
linear function
A linear function is a polynomial function of degree one, which means it has the form \( y = mx + b \).
Here, \( m \) is the slope of the line, and \( b \) is the y-intercept.
In mathematical terms: \[ y = mx + b \]
Linear functions have a constant rate of change, characterized by the slope \( m \).
In our exercise, the given function \( y = \frac{3x}{4} \) can be seen as a linear function where the slope \( m \) is \( \frac{3}{4} \) and the y-intercept \( b \) is zero.
This means the function is a straight line passing through the origin with a consistent slope.
basic differentiation
Basic differentiation involves finding the derivative of simple functions using basic rules.
Key rules include:
  • Power Rule: \[ \frac{d}{dx}(x^n) = nx^{n-1} \]
  • Constant Multiple Rule: \[ \frac{d}{dx}(kx) = k \]
  • Sum Rule: The derivative of a sum is the sum of the derivatives.
For the exercise problem, we use the constant multiple rule.
The differentiation process is simplified, as we're dealing with a basic linear function.
This makes it unnecessary to apply more complicated rules or techniques.
Consequently, the derivative \( \frac{d}{dx} \left( \frac{3}{4} x \right) \) equals \( \frac{3}{4} \).
Mastering these basic rules will prepare you to tackle more complex differentiation problems in calculus.

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. $$y=\frac{x^{5}+x}{x^{2}}$$

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

For each function, find the interval(s) for which \(f^{\prime}(x)\) is positive. $$f(x)=\frac{1}{3} x^{3}-x^{2}-3 x+5$$

For each function, find the interval(s) for which \(f^{\prime}(x)\) is positive. $$f(x)=x^{2}-4 x+1$$

Find \(d y / d x .\) Each function can be differentiated using the rules developed in this section, but some algebra may be required beforehand. $$y=\frac{5 x^{2}-8 x+3}{8}$$
See all solutions

Recommended explanations on Math Textbooks

Discrete Mathematics

Read Explanation

Logic and Functions

Read Explanation

Probability and Statistics

Read Explanation

Calculus

Read Explanation

Mechanics Maths

Read Explanation

Applied Mathematics

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.