/*! 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 41 Find \(d y\) $$y=\sqrt{x+1}$$... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 2: Problem 41

Find \(d y\) $$y=\sqrt{x+1}$$

Short Answer

Expert verified
dy = \frac{1}{2} \frac{dx}{\sqrt{x + 1}}

Step by step solution

01

Identify the function

The given function is \( y = \sqrt{x+1} \).
02

Recall the chain rule

To differentiate a composite function, the chain rule is used. The chain rule states \( d y = \frac{dy}{dx}dx \).
03

Differentiate the outer function

The outer function is the square root function. Its derivative is given by \( \frac{d}{du} \sqrt{u} = \frac{1}{2\sqrt{u}} \).
04

Differentiate the inner function

The inner function is \( x + 1 \). The derivative of \( x + 1 \) is \( \frac{d}{dx}(x + 1) = 1 \).
05

Apply the chain rule

Using the chain rule, \( dy = \frac{d}{dx}(\sqrt{x+1})dx = \frac{1}{2\sqrt{x+1}} \times 1 dx = \frac{1}{2\sqrt{x+1}} dx \).

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.

Composite Function
A composite function is a function made up of two or more functions. It essentially means you're putting one function inside another. For example, in the given exercise, we have the function \( y = \sqrt{x+1} \). Here, \( \sqrt{u} \) is the outer function, and \( x + 1 \) is the inner function.
  • The outer function operates on the result of the inner function.
  • This composition of functions requires a special rule for differentiation.

Understanding how to identify and work with composite functions is essential. You can solve more complex differentiation problems with the chain rule by mastering this concept.
Outer Function Derivative
In the context of the chain rule, we first need to identify the outer function, which is the function that 'holds' the inner function.
  • In the given problem, \( y = \sqrt{x+1} \), the outer function is \( \sqrt{u} \), where \( u = x + 1 \).
  • The derivative of this outer function \( \sqrt{u} \) is \( \frac{1}{2\sqrt{u}} \).

We use this derivative because it tells us how the outer function changes in response to changes in the inner function. This step is crucial because it lays the groundwork for correctly applying the chain rule.
Inner Function Derivative
After differentiating the outer function, our next focus is on the inner function. This step involves finding out how the inner function changes with respect to the original variable, typically \( x \).
  • For the given function \( y = \sqrt{x+1} \), the inner function is \( x + 1 \).
  • The derivative of \( x + 1 \) with respect to \( x \) is simply 1, which means \( \frac{d}{dx}(x + 1) = 1 \).

Once we have the derivatives of both the outer and inner functions, we can finally apply the chain rule to find the derivative of the composite function. This step ensures that every aspect of the composite function's variation is accounted for correctly.

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

Find the absolute maximum and minimum values of the function, if they exist, over the indicated interval. $$g(x)=x \sqrt{x+3} ; \quad[-3,3]$$

Graph cach function. Then estimate any relative extrema. Where appropriate, round to three dreimal places. $$f(x)=4 x-6 x^{2 / 3}$$

Graph cach function. Then estimate any relative extrema. Where appropriate, round to three dreimal places. $$f(x)=x-\sqrt{x}$$

Find \(d y\) $$y=x^{4}-2 x^{3}+5 x^{2}+3 x-4$$

Sketch the graph of each function. List the coordinates of where extrema or points of inflection occur. State where the function is increasing or decreasing, as well as where it is concave up or concave down. f(x)=-x^{3}+3 x-2
See all solutions

Recommended explanations on Math Textbooks

Statistics

Read Explanation

Discrete Mathematics

Read Explanation

Mechanics Maths

Read Explanation

Applied Mathematics

Read Explanation

Probability and Statistics

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.