/*! 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 If \(y=\log \left(x+\sqrt{x^{2}+... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 6: Problem 18

If \(y=\log \left(x+\sqrt{x^{2}+1}\right)\), find \(\frac{d y}{d x}\)

Short Answer

Expert verified
\(\frac{d y}{d x} = \frac{1}{x + \sqrt{x^2 + 1}} * \left(1 + \frac{x}{\sqrt{x^2 + 1}}\right)\)

Step by step solution

01

Identify the outer function

The outer function here is \( f(x) = \log x \). So, starting by finding the derivative of the outer function, got \( f'(x) = \frac{1}{x} \)
02

Identify the inner function

The inner function here is \( g(x) = x + \sqrt{x^2 + 1} \). We must find its derivative.
03

Apply chain rule and find derivative of the inner function

To find \( g'(x) \), first differentiate each term separately. The derivative of \( x \) is \( 1 \). The derivative of \( \sqrt{x^2 + 1} \) is \( \frac{x}{\sqrt{x^2 + 1}} \) according to the chain rule after substituting \( u = x^2 + 1 \) . So, \( g'(x) = 1 + \frac{x}{\sqrt{x^2 + 1}} \).
04

Combine using the chain rule

According to the chain rule, the derivative of the composite function \( y = f(g(x)) \) is \( f'(g(x))*g'(x) \). Substituting in the values from Steps 1 and 3, got \( \frac{d y}{d x} = \frac{1}{x + \sqrt{x^2 + 1}} * \left(1 + \frac{x}{\sqrt{x^2 + 1}}\right) \)

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 \(y=\tan ^{-1}\left\\{\frac{\sqrt{1+x^{2}}+\sqrt{1-x^{2}}}{\sqrt{1+x^{2}}-\sqrt{1-x^{2}}}\right\\}\), find \(\frac{d y}{d x}\).

If \(x=a(1-\cos \theta), y=a(\theta+\sin \theta)\), prove that \(\frac{d^{2} y}{d x^{2}}=-\frac{1}{a}\) at \(\theta=\frac{\pi}{2}\)

If \(\sqrt{x^{2}+y^{2}}=a e^{\tan ^{-1}} x\), where \(a>0, y(0) \neq 0\) then find the value of \(y^{\prime \prime}(0)\).

If \(y=\tan ^{-1}\left(\frac{\sqrt{1+x}-\sqrt{1-x}}{\sqrt{1+x}+\sqrt{1-x}}\right)\), find \(\frac{d y}{d x} .\)

If \(y=\tan ^{-1} x\), prove that \(\left(1+x^{2}\right) y_{2}+2 x y_{1}=0\)
See all solutions

Recommended explanations on Math Textbooks

Mechanics Maths

Read Explanation

Logic and Functions

Read Explanation

Pure Maths

Read Explanation

Applied Mathematics

Read Explanation

Decision Maths

Read Explanation

Probability and 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.