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Chapter 3: Problem 8

Give three different notations for the derivative of \(f\) with respect to \(x\)

Short Answer

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Answer: The three different notations for representing the derivative of a function (f) with respect to a variable (x) are: 1) Lagrange's Notation: \(f'(x)\), 2) Leibniz's Notation: \(\frac{df}{dx}\), and 3) Newton's Notation: \(\dot{f}(x)\) or \(\ddot{f}(x)\).

Step by step solution

01

Notation 1: Lagrange's Notation

In Lagrange's notation, the derivative of the function \(f\) with respect to \(x\) is denoted as \(f'(x)\). This is a very common notation in calculus and indicates that we are considering the first derivative of the function \(f\) with respect to the variable \(x\).
02

Notation 2: Leibniz's Notation

In Leibniz's notation, the derivative of the function \(f\) with respect to \(x\) is denoted as \(\frac{df}{dx}\). This notation is also commonly used in calculus and emphasizes the relationship between the change in the function (\(df\)) and the change in the variable (\(dx\)).
03

Notation 3: Newton's Notation

In Newton's notation, the derivative of the function \(f\) with respect to \(x\) is denoted as \(\dot{f}(x)\) or \(\ddot{f}(x)\). However, these notations are less common nowadays and are mainly used in physics, specifically in the context of time derivatives and acceleration.

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Key Concepts

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

Lagrange's Notation
Lagrange's Notation is one of the most straightforward ways to express the derivative of a function. In this form, the derivative of a function \(f\) with respect to \(x\) is written as \(f'(x)\). This symbolically represents the rate of change or the slope of the function \(f\) at any given point \(x\). It鈥檚 a concise way to remind us we're looking at the first derivative.

A few key things to remember about Lagrange's notation include:
  • It's typically used when discussing the first derivative, although higher-order derivatives are noted with more primes, like \(f''(x)\) for the second derivative.
  • This notation is prevalent in calculus and often used in introductory calculus courses, as it's visually simple and easy to read.
Lagrange鈥檚 notation serves as a shorthand that is practical in both teaching and solving calculus problems. It reminds us of the function in question while keeping the notation uncluttered.
Leibniz's Notation
Leibniz's Notation is another classical way to denote the derivative, offering a slightly different perspective than Lagrange鈥檚. In this form, you write the derivative of a function \(f\) about \(x\) as \(\frac{df}{dx}\). This method highlights the idea of infinitesimal changes: the numerator \(df\) stands for the small change in \(f\), and the denominator \(dx\) represents the small change in \(x\).

There are several advantages to using Leibniz鈥檚 notation:
  • It gives a clear idea of what variables we are differentiating concerning. For instance, \(\frac{dy}{dx}\) indicates that \(y\) is differentiated with respect to \(x\).
  • It serves as a useful tool in integration and differentiation techniques, such as the chain rule or integration by substitution.
Leibniz's notation is widely used because it efficiently communicates the relationship between changes in different variables, which is fundamental to calculus.
Newton's Notation
Newton's Notation brings a unique approach to expressing derivatives, typically represented in forms \(\dot{f}(x)\) and \(\ddot{f}(x)\). Although less prevalent in modern calculus textbooks, it holds particular importance in the fields of physics and engineering.

Here is what you should know about Newton鈥檚 notation:
  • It is mainly used to denote derivatives with respect to time. For example, \(\dot{f}(x)\) represents velocity when \(f(x)\) is position, and \(\ddot{f}(x)\) stands for acceleration.
  • This notation doesn鈥檛 explicitly show what variable the function is differentiated concerning, which is why it is mainly used when time is the variable being considered.
While not as common in calculus, Newton鈥檚 notation is indispensable in differential equations and physics, where time derivatives are frequently used.

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Most popular questions from this chapter

Complete the following steps. a. Find equations of all lines tangent to the curve at the given value of \(x\). b. Graph the tangent lines on the given graph. \(4 x^{3}=y^{2}(4-x) ; x=2\) (cissoid of Diocles)

a. Determine an equation of the tangent line and normal line at the given point \(\left(x_{0}, y_{0}\right)\) on the following curves. b. Graph the tangent and normal lines on the given graph. $$\begin{aligned} &3 x^{3}+7 y^{3}=10 y\\\ &\left(x_{0}, y_{0}\right)=(1,1) \end{aligned}$$

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The flow of a small stream is monitored for 90 days between May 1 and August 1. The total water that flows past a gauging station is given by $$V(t)=\left\\{\begin{array}{ll}\frac{4}{5} t^{2} & \text { if } 0 \leq t<45 \\\\-\frac{4}{5}\left(t^{2}-180 t+4050\right) & \text { if } 45 \leq t<90, \end{array}\right.$$ where \(V\) is measured in cubic feet and \(t\) is measured in days, with \(t=0\) corresponding to May 1. a. Graph the volume function. b. Find the flow rate function \(V^{\prime}(t)\) and graph it. What are the units of the flow rate? c. Describe the flow of the stream over the 3 -month period. Specifically, when is the flow rate a maximum?
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