/*! 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 5 Differentiate. $$y=\frac{x}{2}... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 1: Problem 5

Differentiate. $$y=\frac{x}{2}-\frac{2}{x}$$

Short Answer

Expert verified
y' = \frac{1}{2} + 2x^{-2}

Step by step solution

01

- Rewrite the function

Rewrite the function in a form that is easier to differentiate: \[y = \frac{x}{2} - \frac{2}{x} = \frac{x}{2} - 2x^{-1}\]
02

- Apply the power rule

Differentiate each term separately using the power rule: The derivative of \(\frac{x}{2}\) is \( \frac{1}{2} \). The derivative of \-2x^{-1}\ is \2x^{-2}\. So, \[ \frac{d}{dx} \frac{x}{2} = \frac{1}{2} \] \[ \frac{d}{dx} (-2x^{-1}) = 2x^{-2} \]
03

- Combine the results

Combine the results of the differentiated terms: \[ y' = \frac{1}{2} + 2x^{-2} \]

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!

Key Concepts

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

Power Rule
One of the most fundamental rules in differentiation is the Power Rule. It makes finding the derivative of functions involving powers of x straightforward.
The Power Rule states that if you have a function of the form \(f(x) = x^n\), then its derivative is given by:
\[f'(x) = nx^{n-1}\]
This rule applies to any real number exponent, including negative and fractional exponents. For example:
- The derivative of \(x^3\) is \(3x^2\).
- The derivative of \(x^{-2}\) is \(-2x^{-3}\).
Using this rule helps simplify the differentiation process because you can directly apply the formula instead of rederiving it each time.
Derivative
A derivative represents how a function changes as its input changes. Specifically, it measures the rate at which the function's value changes at any given point. In mathematical terms, if you have a function \(y = f(x)\), the derivative \(f'(x)\) tells you the slope of the tangent line to the function's graph at any point x.
Some key points about derivatives include:
  • It helps find instantaneous rates of change. For example, velocity is the derivative of position with respect to time.
  • It allows us to determine the slope of the graph of a function at a specific point.
  • It is foundational in optimization problems, where we look for maximum and minimum values of functions.
To differentiate a function, we often use established rules, like the Power Rule, to simplify the process.
Function Rewriting
Before differentiating a function, it can often be helpful to rewrite it in a form that is easier to work with. This process is known as function rewriting. In the given exercise,
the function is:
\[y = \frac{x}{2} - \frac{2}{x}\]
Differentiating this directly can be cumbersome, so we rewrite it:
\[y = \frac{x}{2} - 2x^{-1}\]
By expressing \(\frac{2}{x}\) as \(2x^{-1}\), we make the power of x explicit, which allows us to use the Power Rule easily. Function rewriting often involves:
  • Changing fractions to negative exponents.
  • Simplifying complex expressions.
  • Using algebraic manipulations to bring the function to a more manageable form.
This step is crucial for simplifying differentiation and making the application of rules like the Power Rule seamless.

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

Examine the graph of the function and evaluate the function-atlarge values of \(x\) to guess the value of the limit. $$\lim _{x \rightarrow \infty} \frac{-8 x^{2}+1}{x^{2}+1}$$

If \(g(3)=2\) and \(g^{\prime}(3)=4,\) find \(f(3)\) and \(f^{\prime}(3),\) where \(f(x)=2 \cdot[g(x)]^{3}\).

Compute the following limits. $$\lim _{x \rightarrow \infty} \frac{5 x+3}{3 x-2}$$

Determine whether each of the following functions is continuous and/or differentiable at \(x=1.\) $$f(x)=\left\\{\begin{array}{ll} x & \text { for } x \neq 1 \\ 2 & \text { for } x=1 \end{array}\right.$$

Let \(y\) denote the percentage of the world population that is urban \(x\) years after 2014. According to data from the United Nations, 54 percent of the world's population was urban in 2014 , and projections show that this percentage will increase to 66 percent by 2050. Assume that \(y\) is a linear function of \(x\) since 2014. (a) Determine \(y\) as a function of \(x.\) (b) Interpret the slope as a rate of change. (c) Find the percentage of the world's population that is urban in 2020. (d) Determine the year in which \(72 \%\) of the world's population will be urban.
See all solutions

Recommended explanations on Math Textbooks

Geometry

Read Explanation

Mechanics Maths

Read Explanation

Statistics

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Calculus

Read Explanation

Discrete Mathematics

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.