/*! 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 58 Find the derivative of the funct... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 5: Problem 58

Find the derivative of the function. \(y=\arctan \frac{x}{2}-\frac{1}{2\left(x^{2}+4\right)}\)

Short Answer

Expert verified
The derivative of \(y=\arctan \frac{x}{2}-\frac{1}{2\left(x^{2}+4\right)}\) is \(y' = \frac{1}{2+x^{2}}-\frac{x}{(x^{2}+4)^2}\).

Step by step solution

01

Step 1. Apply the Chain Rule to arc tangent function

The derivative of \(y = \arctan(u)\) is \(y' = \frac{u'}{1+u^{2}}\) where \(u = \frac{x}{2}\) and \(u' = \frac{1}{2}\). Substituting these values gives: \(\frac{1}{2(1+(\frac{x}{2})^{2})}\)
02

Step 2. Simplify the expression

The simplified form is \(\frac{1}{2+x^{2}}\).
03

Step 3. Apply the Chain Rule to the fraction

The derivative of \(y = v^{-1}\) where \(-1\) is a power is \(y' = -v^{-2}v'\). Identify \(v = 2(x^{2}+4)\) and \(v' = 4x\). Substitute to get \(-\frac{4x}{4(x^{2}+4)^{2}}\).
04

Step 4. Simplify the expression

This simplifies to \(-\frac{x}{(x^{2}+4)^2}\).
05

Step 5. Add both parts

Finally, sum results from Step 2 and Step 4 which gives \(\frac{1}{2+x^{2}}-\frac{x}{(x^{2}+4)^2}\).

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

Deriving an Inequality Given \(e^{x} \geq 1\) for \(x \geq 0,\) it follows that $$ \int_{0}^{x} e^{t} d t \geq \int_{0}^{x} 1 d t $$ Perform this integration to derive the inequality $$ \begin{array}{l}{e^{x} \geq 1+x} \\ {\text { for } x \geq 0}\end{array} $$

Find an equation of the tangent line to the graph of the function at the given point. \(y=\frac{1}{2} \arccos x, \quad\left(-\frac{\sqrt{2}}{2}, \frac{3 \pi}{8}\right)\)

In Exercises 75鈥�82, find the indefinite integral using the formulas from Theorem 5.20. Exercises 75鈥�82, find the indefinite integral using the formulas from Theorem 5.20. $$ \int \frac{1}{\sqrt{1+e^{2 x}}} d x $$

Find the derivative of the function. \(f(x)=\operatorname{arcsec} 2 x\)

Chemical Reactions Chemicals \(A\) and \(B\) combine in a 3 -to- 1 ratio to form a compound. The amount of compound \(x\) being produced at any time \(t\) is proportional to the unchanged amounts of \(A\) and \(B\) remaining in the solution. So, when 3 kilograms of \(A\) is mixed with 2 kilograms of \(B\) , you have $$\frac{d x}{d t}=k\left(3-\frac{3 x}{4}\right)\left(2-\frac{x}{4}\right)=\frac{3 k}{16}\left(x^{2}-12 x+32\right)$$ One kilogram of the compound is formed after 10 minutes. Find the amount formed after 20 minutes by solving the equation $$\int \frac{3 k}{16} d t=\int \frac{d x}{x^{2}-12 x+32}$$
See all solutions

Recommended explanations on Math Textbooks

Discrete Mathematics

Read Explanation

Pure Maths

Read Explanation

Decision Maths

Read Explanation

Mechanics Maths

Read Explanation

Calculus

Read Explanation

Statistics

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.