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Chapter 3: Problem 34

In Exercises, find \(d y / d x\). $$ y=x+1 / x $$

Short Answer

Expert verified
\( \frac{dy}{dx} = 1 - \frac{1}{x^2} \)

Step by step solution

01

Understand the Expression

The expression given is \( y = x + \frac{1}{x} \). This expression is a combination of a linear term \( x \) and a reciprocal term \( \frac{1}{x} \). Our goal is to differentiate it with respect to \( x \).
02

Differentiate the Linear Term

The first term \( x \) is a simple linear term. The derivative of \( x \) with respect to \( x \) is 1, expressed as \( \frac{d}{dx}(x) = 1 \).
03

Differentiate the Reciprocal Term

Next, we take the derivative of \( \frac{1}{x} \). Recall that the derivative of \( \frac{1}{x} \) is \( -\frac{1}{x^2} \), which can be obtained using the power rule: \( \frac{d}{dx}(x^{-1}) = -x^{-2} \).
04

Combine the Derivatives

Now, add the derivatives of the linear term and the reciprocal term together to find \( \frac{dy}{dx} \). This results in \( \frac{dy}{dx} = 1 - \frac{1}{x^2} \).

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Key Concepts

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

Derivative Calculation
Differentiation is a fundamental concept in calculus that describes how a function changes as its input changes. It's like finding the slope of the function's graph at any given point.
  • The original function we are working with is: \[ y = x + \frac{1}{x} \]
  • When we differentiate to find \( \frac{dy}{dx} \), we separate into parts and calculate each term independently.
The process involves taking known derivative rules and applying them to each term:
  • The derivative of \( x \) is simply 1. This result stems from the fact that when you graph \( y = x \), the slope at any point is constant, equal to 1.
  • The derivative of \( \frac{1}{x} \) uses the power rule applied to the expression \( x^{-1} \). This rule states that \( \frac{d}{dx}(x^n) = nx^{n-1} \), leading to \(-x^{-2}\) or \(-\frac{1}{x^2}\).
After handling each part, you then combine these derivatives to form the final derivative of the entire expression.
Linear Term
A linear term in any expression refers to a term of the first degree, typically taking the form \( ax + b \). These terms are straightforward to differentiate because their rate of change, or slope, is constant.In the given exercise, the linear term is simply \( x \). When dealing with a basic linear function such as this:
  • Derivative of \( x \) with respect to \( x \) is a constant 1. This result comes from the direct relation in its slope, meaning the rate of change is unchanging.
The simplicity of linear functions, like \( y = x \), makes their calculations easy to predict and serves as the foundational building block for understanding more complex derivatives.
Reciprocal Term
A reciprocal term is one where the variable is in the denominator, such as \( \frac{1}{x} \). These terms introduce a bit more complexity due to how division by a variable affects the overall rate of change. The derivative of a reciprocal function, like \( \frac{1}{x} \), is derived using the power rule:
  • Write \( \frac{1}{x} \) as \( x^{-1} \). This transformation allows us to use the power rule for differentiation.
  • According to the power rule, differentiate \( x^{-1} \) to obtain \( -x^{-2} \), or rephrased as \(-\frac{1}{x^2}\).
This negative sign indicates that as \( x \) increases, \( \frac{1}{x} \) decreases at a decreasing rate, representing the characteristic slipping slope needed for further mathematical analysis.

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Most popular questions from this chapter

Use the results of this section to find the derivative of the given function at the given numbers. $$ f(x)=x^{3} ; a=-\frac{1}{4} $$

The distance \(s\) traveled by a moving particle is given by \(s=a e^{k t}+b e^{-k t}\), where \(a, b\), and \(k\) are positive constants and \(t\) denotes time. Show that the acceleration of the particle is proportional to the distance traveled.

A baseball player chasing a fly ball runs in a straight line toward the right field fence. Set up a coordinate system, with feet as units, such that the \(y\) axis represents the fence and the player runs along the negative \(x\) axis toward the origin. Suppose the player's velocity in feet per second is $$ v(x)=\frac{1}{60} x^{2}+\frac{11}{10} x+25 \quad \text { for }-30 \leq x \leq 0 $$ when the player is located at \(x .\) What is the acceleration when the player is 1 foot from the fence? (Hint: Use the Chain Rule.)

Find the derivative of the given function. $$ f(t)=7 t^{-5} $$

Use the results of this section to find the derivative of the given function at the given numbers. $$ f(t)=\sin t ; a=\pi / 4, \pi / 3 $$
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