By computing the first few derivatives and looking for a patte...

Question

By computing the first few derivatives and looking for a pattern, find the following derivatives.

b. d¹¹⁰/dx¹¹⁰ (sin x − 3 cos x)

Accepted Answer

Welcome back, everyone. In this problem, by examining the pattern of derivatives, determine the 218th derivative of 5 X + 2 X. Now, if we're going to figure out the 218th derivative, let's try to differentiate 5 multiplied by sin X + 2 cost X a few times to observe the pattern, OK? Now, Let's start by getting the first derivative. So if we let, let's say we let F of X be equal to 5 x x. Plus 2 X. OK. Then that means the first derivative is going to be equal to. The derivative of 5 X X, which is 5 X plus 2 multiplied by the derivative of cos X. So that's 2 multiplied by negative sin X. And that is gonna be equal to 5 + X minus 2 X. Next, Or a second derivative. It is going to be equal to. The derivative of our first derivative and the derivative of 5 X will be -5 X and the derivative of -2 X X will be negative to X. Next, to find our third derivative, we differentiate again. The derivative of -5 sin X is -5 + X, and the derivative of -2 + X is 2 X. OK? And then if we go to our 4th derivative, OK. Then our derivative is going to be 5 x x plus 2 + X. Now let's take a minute and stop here. What do you notice? Notice that by no the 4th derivative is equal to our first derivative 5s function, sorry, 5 x + 2 + X. So this pattern shows that the original function repeats every 4th derivative. Thus this tells us then that the nth derivative will cycle through these four forms depending on the value. That's OK, depending on the value. Of N modulo 4 ND 4. OK. This is very important. In other words, what this is telling us is that if our NA 4 is 1, then it will be the first derivative. If it's 2, it will be the second. If it's 3, it will be the 3rd. And if it is 4, OK, then that means it's gonna be 0 and that will carry us back to our original function. So in this case, if we want to find a 218th derivative, then we need to find 218 mod 4. Which is equal to 2. This tells us that the 218th derivative will have the same form as the second derivative in our pattern, and our second derivative is -5 X minus 2 + X, OK? Therefore, we can say that the 218th derivative. Is -5 x minus 2 X. Thanks a lot for watching everyone. I hope this video helped.

By computing the first few derivatives and looking for a pattern, find the following derivatives.

b. d¹¹⁰/dx¹¹⁰ (sin x − 3 cos x)

Key Concepts

Higher Order Derivatives

Trigonometric Derivatives

Pattern Recognition in Derivatives

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