/*! 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 18 Find \(d y / d x\) by implicit d... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 2: Problem 18

Find \(d y / d x\) by implicit differentiation. $$ x=\sec \frac{1}{y} $$

Short Answer

Expert verified
\( dy/dx = -y^2\sec \left( \frac{1}{y} \right) \cdot \tan \left( \frac{1}{y} \right) \)

Step by step solution

01

Differentiate both sides with respect to x

On the left side, the derivative of x with respect to x is just 1. On the right side, the derivative of \( \sec \frac{1}{y} \) with respect to x is more complex. Recognise that this function can be considered a 'function within a function', that's why we use the chain rule. So firstly, differentiate \( \sec(u) \) where \(u = \frac{1}{y}\) to get \( \sec(u)\tan(u) \). Secondly, differentiate \( u = \frac{1}{y} \) to get \( -\frac{1}{y^2} \). Multiply these two results by the chain rule. So we have: \[ 1 = \sec \left( \frac{1}{y} \right) \cdot \tan \left( \frac{1}{y} \right) \cdot -\frac{1}{y^{2}} \]
02

Solve for dy/dx

Now, rearrange the equation to solve for dy/dx. To isolate dy/dx on one side, multiply the entire equation by -y^2. \[ -y^2 = \sec \left( \frac{1}{y} \right) \cdot \tan \left( \frac{1}{y} \right) \] There you have it! You've found \( dy/dx \) as a function of x and y, using implicit differentiation.

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

If the annual rate of inflation averages \(5 \%\) over the next 10 years, the approximate cost \(C\) of goods or services during any year in that decade is \(C(t)=P(1.05)^{t},\) where \(t\) is the time in years and \(P\) is the present cost. (a) If the price of an oil change for your car is presently \(\$ 24.95,\) estimate the price 10 years from now. (b) Find the rate of change of \(C\) with respect to \(t\) when \(t=1\) and \(t=8\) (c) Verify that the rate of change of \(C\) is proportional to \(C\). What is the constant of proportionality?

Consider the function \(f(x)=\sin \beta x\), where \(\beta\) is a constant. (a) Find the first-, second-, third-, and fourth-order derivatives of the function. (b) Verify that the function and its second derivative satisfy the equation \(f^{\prime \prime}(x)+\beta^{2} f(x)=0\) (c) Use the results in part (a) to write general rules for the even- and odd- order derivatives \(f^{(2 k)}(x)\) and \(f^{(2 k-1)}(x)\) [Hint: \((-1)^{k}\) is positive if \(k\) is even and negative if \(k\) is odd.]

True or False? In Exercises 137-139, determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. If \(y\) is a differentiable function of \(u, u\) is a differentiable function of \(v,\) and \(v\) is a differentiable function of \(x,\) then \(\frac{d y}{d x}=\frac{d y}{d u} \frac{d u}{d v} \frac{d v}{d x}\)

In Exercises 35 and 36, find an equation of the tangent line to the graph of the equation at the given point. $$ \arcsin x+\arcsin y=\frac{\pi}{2}, \quad\left(\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2}\right) $$

Adiabatic Expansion When a certain polyatomic gas undergoes adiabatic expansion, its pressure \(p\) and volume \(V\) satisfy the equation \(p V^{1.3}=k\), where \(k\) is a constant. Find the relationship between the related rates \(d p / d t\) and \(d V / d t\).
See all solutions

Recommended explanations on Math Textbooks

Applied Mathematics

Read Explanation

Probability and Statistics

Read Explanation

Geometry

Read Explanation

Statistics

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.