/*! 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 80 Find \(F^{\prime}(x)\). $$ F... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 4: Problem 80

Find \(F^{\prime}(x)\). $$ F(x)=\int_{-x}^{x} t^{3} d t $$

Short Answer

Expert verified
The derivative \( F^{\prime}(x) \) is \( 0 \).

Step by step solution

01

Understand the Leibniz Rule

According to the Leibniz Rule, the derivative of an integral of a function with variable limits is given by subtracting the function at the lower limit multiplied by the derivative of the lower limit from the function at the upper limit multiplied by the derivative of the upper limit.
02

Apply the Leibniz Rule

Having defined \( F(x) = \int_{-x}^{x} t^{3} dt \), let's apply the Leibniz Rule. We plug in upper limit \( x \) into \( t^3 \), which is \( x^3 \), and multiply by the derivative of upper limit \( x \), which is \(1\). Then, we plug in lower limit \( -x \) into \( t^3 \), which is \( (-x)^3 = -x^3 \), and multiply by the derivative of the lower limit \( -x \), which is \(-1\). Subtract the second term from the first term.
03

Simplify the expression

After applying the rule, the obtained result is \( x^3*1 - (-x^3)*(-1) = x^3 - x^3 = 0 \). Therefore, the derivative of the function \( F(x) \) is \( 0 \).

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

Find \(F^{\prime}(x)\). $$ F(x)=\int_{2}^{x^{2}} \frac{1}{t^{3}} d t $$

Show that \(\arctan (\sinh x)=\arcsin (\tanh x)\).

Find all the continuous positive functions \(f(x),\) for \(0 \leq x \leq\) such that \(\int_{0}^{1} f(x) d x=1, \int_{0}^{1} f(x) x d x=\alpha,\) and \(\int_{0}^{1} f(x) x^{2} d x=\alpha^{2}\) where \(\alpha\) is a real number

Find the limit. \(\lim _{x \rightarrow 0^{-}} \operatorname{coth} x\)

(a) Show that \(\int_{0}^{1} \frac{4}{1+x^{2}} d x=\pi\). (b) Approximate the number \(\pi\) using Simpson's Rule (with \(n=6\) ) and the integral in part (a). (c) Approximate the number \(\pi\) by using the integration capabilities of a graphing utility.
See all solutions

Recommended explanations on Math Textbooks

Statistics

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Applied Mathematics

Read Explanation

Logic and Functions

Read Explanation

Geometry

Read Explanation

Decision 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.