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Problem

Find the 1000th derivative of $ f(x) = xe^{-x}. $

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Problem 77 Hard Difficulty

Find the 50th derivative of $ y = \cos 2x. $


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01:38

Frank Lin

Related Courses

Calculus 1 / AB

Calculus: Early Transcendentals

Chapter 3

Differentiation Rules

Section 4

The Chain Rule

Related Topics

Derivatives

Differentiation

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Senthil K.

August 5, 2021

find the n th derivative of cos2x

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Video Thumbnail

04:40

Derivatives - Intro

In mathematics, a derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much one quantity is changing in response to changes in some other quantity; for example, the derivative of the position of a moving object with respect to time is the object's velocity. The concept of a derivative developed as a way to measure the steepness of a curve; the concept was ultimately generalized and now "derivative" is often used to refer to the relationship between two variables, independent and dependent, and to various related notions, such as the differential.

Video Thumbnail

44:57

Differentiation Rules - Overview

In mathematics, a differentiation rule is a rule for computing the derivative of a function in one variable. Many differentiation rules can be expressed as a product rule.

Join Course
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Problem 16
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Problem 48
Problem 49
Problem 50
Problem 51
Problem 52
Problem 53
Problem 54
Problem 55
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Problem 99
Problem 100

Video Transcript

The goal in this problem is to find the 50th derivative of why. But obviously we're not going to find 50 derivatives. We're going to find a few and look for a pattern and then generalize it. So let's start by finding the first derivative using the chain rule. The derivative of co sign is negative sign. So we have negative sighing of two x times, the derivative of two x two. So this is negative to sign of two X. Now let's find the derivative of that and that will be the second derivative. So let's keep the negative to the constant. And then let's take the derivative of Sign, which is co signs with co sign of two X times. The derivative of the inside the derivative of two X is too and will simplify that, and we have negative four co sign of two X. Now let's find the drift of of that and that will be why Triple prime. So we'll leave the constant negative four and then the derivative of co sign is negative sign. So have negative sign of two x times the derivative of the inside, too. So now we have eight sign of two X. Now let's find the fourth derivative so the derivative will leave the eight and then the derivative of Sign of two X would be co sign of two x times, the derivative of two x two. So now we have 16 co sign of two X. Now let's find a pattern here. So because we're back to co sign of two X, we're kind of like back to the beginning, and the next one would have a negative sign and the next one would have a negative co sign the next one having a positive sign. The next one have a positive co sign. So have a pattern that goes every four. And then we also have something going on with the numbers. So notice we have to to the 1st 22 to the 2nd 42 to the 3rd 8 and two to the 4th 16 Okay, so what we could expect, Let's say if we were jumping ahead to the eighth derivative, we could expect it to have a two to the eighth and a co sign of two X. And if we're jumping ahead to this 12th derivative because we're going in groups of four. We would expect it to have a two to the 12th times a co sign of two X. Nellis jumped way ahead. Suppose we were looking at the 48th derivative. We would expect that to have a two of the 48 power times a co sign of two X. So from here, let's figure out the 49th and 50th. Okay, So if we're if we're at the 48th derivative and we're here in the pattern, then we're gonna go back up to here for the 49th and then to hear for the 50th which means we're going to have a negative. We're going to have to to the 50th and we're going to be at a co sign of two X. So our answer is, the 50th derivative is the opposite of two to the 50th Power Times, a co sign of two x

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Video Thumbnail

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Derivatives - Intro

In mathematics, a derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much one quantity is changing in response to changes in some other quantity; for example, the derivative of the position of a moving object with respect to time is the object's velocity. The concept of a derivative developed as a way to measure the steepness of a curve; the concept was ultimately generalized and now "derivative" is often used to refer to the relationship between two variables, independent and dependent, and to various related notions, such as the differential.

Video Thumbnail

44:57

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