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Chapter 3: Problem 7

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

Short Answer

Expert verified
Answer: The notation \(\frac{dy}{dx}\) is used to represent the derivative because it provides a simple and intuitive representation of the rate of change of one variable with respect to another. It also originates from centuries-old mathematical notation introduced by famous mathematicians, such as Guillaume de l'H么pital and Isaac Newton, and effectively communicates the concept of limits and infinitesimally small changes in quantities, which are central to the concept of calculus in continuous functions.

Step by step solution

01

Understand the meaning of the derivative

The derivative is a measure of how much a function changes as its input changes. In other words, it's the rate of change of a function. In the context of calculus, it is the "slope" or "gradient" of a function and is a central concept that allows us to describe the behavior of functions when they are changing rapidly.
02

Discuss the notation origins

The notation \(\frac{dy}{dx}\) was introduced by the French mathematician Guillaume de l'H么pital and is often attributed to the English mathematician and physicist Sir Isaac Newton as well. The notational format was a convenient way to represent the rate of change in one variable with respect to another variable. During the development of continuous functions and the calculus, mathematicians were trying to find the best way to represent this concept succinctly and understandably for other mathematicians.
03

Understand the meaning of "y" and "x" in \(\frac{dy}{dx}\) notation

In the notation \(\frac{dy}{dx}\), the variable \(y\) typically represents the dependent variable (i.e., the output of the function), whereas the variable \(x\) represents the independent variable (i.e., the input of the function). The differential symbols \(d\) represent infinitesimally small (almost zero) changes in the variables \(x\) and \(y\).
04

Interpret the \(\frac{dy}{dx}\) notation as a limit of the ratio of small changes in \(y\) and \(x\)

When we write the derivative using the notation \(\frac{dy}{dx}\), we are essentially referring to the limit of the ratio when both the change in \(y\) (denoted as \(\Delta{y}\)) and the change in \(x\) (denoted as \(\Delta{x}\)) approach zero. Mathematically, this is expressed using the limit notation: $$ \frac{dy}{dx} = \lim_{\Delta{x} \to 0} \frac{\Delta{y}}{\Delta{x}} $$
05

Summarize the reasons for using the notation

So, to conclude, the notation \(\frac{dy}{dx}\) is used to represent the derivative because: 1. It provides a simple and intuitive representation of the rate of change of one variable with respect to another; 2. It originates from centuries-old mathematical notation introduced by famous mathematicians, such as Guillaume de l'H么pital and Isaac Newton; 3. It effectively communicates the concept of limits and infinitesimally small changes in quantities, which are central to the concept of calculus in continuous functions.

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Most popular questions from this chapter

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. \(x+y^{3}-x y=1\) (Hint: Rewrite as \(y^{3}-1=x y-x\) and then factor both sides.)

Two boats leave a port at the same time, one traveling west at \(20 \mathrm{mi} / \mathrm{hr}\) and the other traveling southwest at \(15 \mathrm{mi} / \mathrm{hr} .\) At what rate is the distance between them changing 30 min after they leave the port?

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. $$y^{2}=\frac{x^{2}(4-x)}{4+x} \text { (right strophoid) }$$

Once Kate's kite reaches a height of \(50 \mathrm{ft}\) (above her hands), it rises no higher but drifts due east in a wind blowing \(5 \mathrm{ft} / \mathrm{s} .\) How fast is the string running through Kate's hands at the moment that she has released \(120 \mathrm{ft}\) of string?

Suppose your graphing calculator has two functions, one called sin \(x,\) which calculates the sine of \(x\) when \(x\) is in radians, and the other called \(s(x),\) which calculates the sine of \(x\) when \(x\) is in degrees. a. Explain why \(s(x)=\sin \left(\frac{\pi}{180} x\right)\) b. Evaluate \(\lim _{x \rightarrow 0} \frac{s(x)}{x} .\) Verify your answer by estimating the limit on your calculator.
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