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Problem 77 Hard Difficulty

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

Answer

50 th derivative of $y=-2^{50} \cos (2 x)$

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SK

Senthil K.

August 5, 2021

find the n th derivative of cos2x

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