/*! 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 64 Find the derivative of the funct... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 7: Problem 64

Find the derivative of the function. State which differentiation rule(s) you used to find the derivative. $$ y=\sqrt{x}(x-2)^{2} $$

Short Answer

Expert verified
The derivative of the function \(y=\sqrt{x}(x-2)^{2}\) is \(y' = \frac{(x-2)^{2}}{2\sqrt{x}} + 2\sqrt{x}(x-2)\). The differentiation rules used here are the Product Rule and the Chain Rule.

Step by step solution

01

Identify the functions and rules

In this exercise, the main function \(y\) is a product of two parts: \(\sqrt{x}\) and \((x-2)^{2}\). To differentiate this, we'll use the product rule. Also, the second part of the function \((x-2)^{2}\) will require the chain rule.
02

Apply the Product Rule

The product rule states that the derivative of a product of two functions is the derivative of the first times the second plus the first times the derivative of the second. Therefore, \(y'\) can be shown as:\(y' = \sqrt{x}'(x-2)^{2}+ \sqrt{x}(x-2)^{2}'\)
03

Differentiate \(\sqrt{x}\)

The derivative of \(\sqrt{x}\) is \(\frac{1}{2\sqrt{x}}\). Therefore we can update the equation:\(y' = \frac{1}{2\sqrt{x}}(x-2)^{2}+ \sqrt{x}(x-2)^{2}'\)
04

Apply the Chain Rule on \((x-2)^{2}\)

The chain rule tells us how to differentiate a composition of functions. It states that the derivative of a composition of functions is the derivative of the outer function times the derivative of the inner function. Applying it to \((x-2)^{2}\), we get: \((x-2)^{2}' = 2(x-2)\times1 = 2(x-2)\). We can now insert this into the equation:\(y' = \frac{1}{2\sqrt{x}}(x-2)^{2}+ \sqrt{x}*2(x-2)\)
05

Simplify the Expression

Let's simplify this. It yields:\(y' = \frac{(x-2)^{2}}{2\sqrt{x}} + 2\sqrt{x}(x-2)\)

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

Use the General Power Rule to find the derivative of the function. $$ y=\sqrt[3]{9 x^{2}+4} $$

Find the derivative of the function. State which differentiation rule(s) you used to find the derivative. $$ f(x)=\frac{1}{\left(x^{2}-3 x\right)^{2}} $$

Find an equation of the tangent line to the graph of the function at the given point. Then use a graphing utility to graph the function and the tangent line in the same viewing window. $$ f(t)=\left(t^{2}-9\right) \sqrt{t+2} \quad(-1,-8) $$

Use the General Power Rule to find the derivative of the function. $$ g(x)=(4-2 x)^{3} $$

Find the derivative of the function. State which differentiation rule(s) you used to find the derivative. $$ y=-\frac{4}{(t+2)^{2}} $$
See all solutions

Recommended explanations on Math Textbooks

Logic and Functions

Read Explanation

Mechanics Maths

Read Explanation

Probability and Statistics

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Decision Maths

Read Explanation

Applied 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.