/*! 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 Why is the notation \(\frac{d y}... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 3: Problem 7

Why is the notation \(\frac{d y}{d x}\) used to represent the derivative?

Short Answer

Expert verified
Question: Explain the notation \(\frac{dy}{dx}\) for representing derivatives and its relationship with the concept of a derivative. Answer: The notation \(\frac{dy}{dx}\) represents the derivative of a function, which indicates the rate of change of the dependent variable (\(y\)) with respect to its independent variable (\(x\)). This notation highlights the idea that the derivative of a function signifies how much \(y\) changes with respect to \(x\), as derived from the limit definition of derivatives. It was introduced by Leibniz, who used differentials \(dx\) and \(dy\) to represent small increments in \(x\) and \(y\), and the derivative represents the ratio of these small increments. Overall, the notation \(\frac{dy}{dx}\) embodies both the conceptual understanding and mathematical principles behind the derivative.

Step by step solution

01

Understanding the concept of derivative

Before explaining the notation, it is important to understand the concept of a derivative. The derivative of a function represents the rate of change of the function with respect to its independent variable. In other words, it represents how quickly the dependent variable (output) changes with respect to the independent variable (input). In this notation, \(x\) is the independent variable and \(y\) is the dependent variable.
02

Derived from the limit definition of derivatives

The notation for the derivative, \(\frac{dy}{dx}\), is derived from the limit definition of derivatives. The limit definition of a derivative is given by: $$f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}.$$ Here, \(f'(x)\) represents the derivative of function \(f\) with respect to \(x\). This definition helps us understand the meaning behind the notation as the limit represents how the change in \(f(x)\) varies with a small change in \(x\).
03

Introduction of differential notation

The notation \(\frac{dy}{dx}\) was introduced by Leibniz, one of the inventors of calculus, who used the notation \(dx\) (differential of \(x\)) and \(dy\) (differential of \(y\)) to express small increments in \(x\) and \(y\). The derivative then represents the ratio of these small increments, i.e., how much does \(y\) change with respect to \(x\)?
04

The meaning behind the notation

The notation \(\frac{dy}{dx}\) can literally be interpreted as the ratio of the infinitesimal change in \(y\) to the infinitesimal change in \(x\). It highlights the idea that the derivative of a function represents the rate of change of its dependent variable (\(y\)) with respect to its independent variable (\(x\)).
05

Summary

In conclusion, the notation \(\frac{dy}{dx}\) is used to represent the derivative because it conveys the notion of the rate of change of the dependent variable (\(y\)) with respect to the independent variable (\(x\)), in a way that is derived from the limit definition of derivatives and consistent with the intuitive idea of change in one variable relative to another.

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!

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

Let \(C(x)\) represent the cost of producing \(x\) items and \(p(x)\) be the sale price per item if \(x\) items are sold. The profit \(P(x)\) of selling x items is \(P(x)=x p(x)-C(x)\) (revenue minus costs). The average profit per item when \(x\) items are sold is \(P(x) / x\) and the marginal profit is dP/dx. The marginal profit approximates the profit obtained by selling one more item given that \(x\) items have already been sold. Consider the following cost functions \(C\) and price functions \(p\). a. Find the profit function \(P\). b. Find the average profit function and marginal profit function. c. Find the average profit and marginal profit if \(x=a\) units are sold. d. Interpret the meaning of the values obtained in part \((c)\). $$\begin{aligned} &C(x)=-0.04 x^{2}+100 x+800, p(x)=200-0.1 x,\\\ &\bar{a}=1000 \end{aligned}$$

Use the properties of logarithms to simplify the following functions before computing \(f^{\prime}(x)\). $$f(x)=\ln \left(\sec ^{4} x \tan ^{2} x\right)$$

Identifying functions from an equation. The following equations implicitly define one or more functions. a. Find \(\frac{d y}{d x}\) using implicit differentiation. b. Solve the given equation for \(y\) to identify the implicitly defined functions \(y=f_{1}(x), y=f_{2}(x), \ldots\) c. Use the functions found in part (b) to graph the given equation. \(y^{3}=a x^{2}(\text { Neile's semicubical parabola })\)

Calculate the derivative of the following functions (i) using the fact that \(b^{x}=e^{x \ln b}\) and (ii) by using logarithmic differentiation. Verify that both answers are the same. $$y=\left(x^{2}+1\right)^{x}$$

Use the properties of logarithms to simplify the following functions before computing \(f^{\prime}(x)\). $$f(x)=\ln \frac{(2 x-1)(x+2)^{3}}{(1-4 x)^{2}}$$
See all solutions

Recommended explanations on Math Textbooks

Applied Mathematics

Read Explanation

Logic and Functions

Read Explanation

Mechanics Maths

Read Explanation

Statistics

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Geometry

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.