Log out Go to App
Go to App

Learning Materials

Features

Discover

Chapter 10: Problem 789

If \(\mathrm{y}=3 \mathrm{x}^{(3 / 2)}(\mathrm{x}-1)\) then \(|[\mathrm{dy} / \mathrm{d} \mathrm{x}]|_{(\mathrm{x}=1)}=\) (a) 6 (b) 3 (c) 1 (d) 0

Short Answer

Expert verified
The absolute value of the derivative of the function \(y=3x^{(3/2)}(x-1)\) evaluated at \(x=1\) is: (b) 3

Step by step solution

01

Identify the function y

The given function is y=3x^(3/2)(x-1).
02

Apply the product rule

The function can be rewritten as the product of two functions u(x) and v(x): u(x) = 3x^(3/2) v(x) = (x-1) We will now apply the product rule in order to find the derivative. The product rule states that if y = u(x)v(x), then the derivative of y with respect to x is: (dy/dx) = (du/dx)v(x) + u(x)(dv/dx)
03

Differentiate u(x) and v(x)

First, let's find the derivatives of u(x) and v(x) with respect to x. Apply the power rule to differentiate u(x): du/dx = d(3x^(3/2))/dx = (3/2) * 3x^(1/2) Now, differentiate v(x) with respect to x: dv/dx = d(x-1)/dx = 1
04

Compute the derivative of y

Now that we have the derivatives of u(x) and v(x), we can compute the derivative dy/dx: dy/dx = (du/dx)v(x) + u(x)(dv/dx) = (3/2) * 3x^(1/2)(x-1) + 3x^(3/2)(1)
05

Evaluate the derivative at x=1

We need to find the value of dy/dx when x=1: (dy/dx)|_(x=1) = (3/2) * 3(1)^(1/2)(1-1) + 3(1)^(3/2)(1) = 0 + 3
06

Find the absolute value

Finally, we need to find the absolute value of the derivative evaluated at x=1: |[(dy/dx)|_(x=1)]| = |3| From the provided options, the correct answer is: (b) 3

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.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

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 \(\mathrm{y}=\mathrm{f}(\mathrm{f}(\mathrm{f}(\mathrm{x})))\) and \(\mathrm{f}(0)=0, \mathrm{f}^{\prime}(0)=1\) then \(\mid[\mathrm{dy} / \mathrm{d} \mathrm{x}]_{\mathrm{x} 0}=\) (a) 0 (b) 1 (c) - 1 (d) 2

If \(y=\sin (x / 2) \mid[1 /\\{\cos (x / 2) \cos x\\}]+[1 /\\{\cos x \cos (3 x / 2)\\}]\) \(+[1 /\\{\cos (3 \mathrm{x} / 2) \cos 2 \mathrm{x}\\}] \mid\) then \((\mathrm{dy} / \mathrm{dx})_{\mathrm{x}=(\pi / 2)}=\) (a) \((3 / 2)\) (b) \((1 / 2)\) (c) \(-1\) (d) 1

In \(\mathrm{a}+\mathrm{b}+\mathrm{c}=0\), than the equation \(3 \mathrm{ax}^{2}+2 \mathrm{bx}+\mathrm{c}=0\) has, in the interval \((0,1)\) (a) at least one root (b) at most one root (c) no root (d) Exactly one root exist.

If function \(\mathrm{f}(\mathrm{x})=\mathrm{ax}^{3}+\mathrm{bx}^{2}+11 \mathrm{x}-6, \mathrm{x} \in[1,3]\) is satisfying Roll's condition and \(c=|2+(1 / \sqrt{3})|\) than \(\mathrm{a}=\) \(\mathrm{b}=\) (a) \(-1,+6\) (b) \(-2,1\) (c) \(-1,(1 / 2)\) (d) \(1,-6\)

In the interval \(|0,(\pi / 4)|\), then function \(\mathrm{f}(\mathrm{x})=\tan ^{-1}(\sin \mathrm{x}+\) \(\cos x)\) is (a) Decreasing (b) Increasing (c) Both (d) Even Function
See all solutions

Recommended explanations on English Textbooks

Text Comparison

Read Explanation

Cues and Conventions

Read Explanation

Graphology

Read Explanation

Free Response Essay

Read Explanation

Email

Read Explanation

Lexis and Semantics

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.

Sign-up for free