/*! 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 32 Use the limit definition to find... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 7: Problem 32

Use the limit definition to find the derivative of the function. $$ f(x)=1-x^{2} $$

Short Answer

Expert verified
The derivative of the function \(f(x) = 1 - x^{2}\) is \(f'(x)=-2x\).

Step by step solution

01

Insert f(x+h) into the limit definition

First, replace the \(x\) in \(f(x)\) with \(x+h\). This results in \(f(x+h) = 1 - (x+h)^{2}\). Now, the limit definition is \(f'(x)= \lim _{h \rightarrow 0} \frac{1 - (x+h)^{2}- (1 - x^{2})}{h}\).
02

Simplify the numerator

Next, simplify the numerator of the fraction inside the limit. The \(1\) and \(-1\) cancel each other out, and expanding \(-(x+h)^{2}\) gives \(-x^{2}-2xh-h^{2}\). This leads to \( f'(x) = \lim _{h \rightarrow 0} \frac{-x^{2}-2xh-h^{2}+x^{2}}{h}\). This further simplifies to \(f'(x)= \lim _{h \rightarrow 0}\frac{-2xh - h^{2}}{h}\).
03

Cancel out h

In this step, cancel out \(h\) from the numerator and the denominator, resulting in \(f'(x)= \lim _{h \rightarrow 0}(-2x - h)\).
04

Take the limit

Lastly, take the limit as \(h\) approaches 0, remaining with \(f'(x)=-2x\).

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. $$ h(x)=\left(6 x-x^{3}\right)^{2} $$

Find \(d y / d u, d u / d x\), and \(d y / d x\). $$ y=u^{3}, u=3 x^{2}-2 $$

Use a symbolic differentiation utility to find the derivative of the function. Graph the function and its derivative in the same viewing window. Describe the behavior of the function when the derivative is zero. $$ f(x)=\sqrt{\frac{x+1}{x}} $$

Find the derivative of the function. State which differentiation rule(s) you used to find the derivative. $$ h(t)=\frac{t+2}{t^{2}+5 t+6} $$

Use the General Power Rule to find the derivative of the function. $$ y=2 \sqrt{4-x^{2}} $$
See all solutions

Recommended explanations on Math Textbooks

Mechanics Maths

Read Explanation

Logic and Functions

Read Explanation

Applied Mathematics

Read Explanation

Probability and Statistics

Read Explanation

Decision Maths

Read Explanation

Pure Maths

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.