/*! 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 51 Find the derivative of the funct... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 7: Problem 51

Find the derivative of the function. State which differentiation rule(s) you used to find the derivative. $$ y=\frac{1}{x-2} $$

Short Answer

Expert verified
The derivative of the function \( y=\frac{1}{x-2} \) is \( y' = -\frac{1}{{(x-2)}^2} \)

Step by step solution

01

Rewrite the Function

Rewrite the function in a form that will make differentiation easier: \( y=(x-2)^{-1} \)
02

Apply the Power Rule

Since the function is now in the form of \( y=(x-2)^{-1} \), we will use the power rule for differentiation, which states that for any real number 'n', the derivative of \( x^n \) with respect to 'x' is \( nx^{n-1} \). Therefore, the derivative of our function \( y=(x-2)^{-1} \) is \( y' = -1*(x-2)^{-2} \)
03

Simplify the Result

Simplifying, we can rewrite \( -1*(x-2)^{-2} \) as \( -\frac{1}{{(x-2)}^2} \)

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

An environmental study indicates that the average daily level \(P\) of a certain pollutant in the air, in parts per million, can be modeled by the equation \(P=0.25 \sqrt{0.5 n^{2}+5 n+25}\) where \(n\) is the number of residents of the community, in thousands. Find the rate at which the level of pollutant is increasing when the population of the community is 12,000 .

Use a symbolic differentiation utility to find the derivative of the function. Graph the function and its derivative in the same viewing window. Describe the behavior of the function when the derivative is zero. $$ f(x)=\sqrt{\frac{x+1}{x}} $$

Given \(f(x)=x+1\), which function would most likely represent a demand function? Explain your reasoning. Use a graphing utility to graph each function, and use each graph as part of your explanation. (a) \(p=f(x)\) (b) \(p=x f(x)\) (c) \(p=-f(x)+5\)

You decide to form a partnership with another business. Your business determines that the demand \(x\) for your product is inversely proportional to the square of the price for \(x \geq 5\). (a) The price is \(\$ 1000\) and the demand is 16 units. Find the demand function. (b) Your partner determines that the product costs \(\$ 250\) per unit and the fixed cost is \(\$ 10,000\). Find the cost function. (c) Find the profit function and use a graphing utility to graph it. From the graph, what price would you negotiate with your partner for this product? Explain your reasoning.

Identify the inside function, \(u=g(x)\), and the outside function, \(y=f(u)\). $$ y=f(g(x)) \quad u=g(x) \quad y=f(u) $$ $$ y=\left(x^{2}+1\right)^{4 / 3} $$
See all solutions

Recommended explanations on Math Textbooks

Probability and Statistics

Read Explanation

Mechanics Maths

Read Explanation

Logic and Functions

Read Explanation

Discrete Mathematics

Read Explanation

Applied Mathematics

Read Explanation

Calculus

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.