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.

c. d⁷³/dx⁷³ (x sin x)

Accepted Answer

Below there today we're going to solve the following practice problem together. So first off, let us read the problem and highlight all the key pieces of information that we need to use in order to solve this problem. By examining the pattern of derivatives, determine the ninety-seventh derivative of 3 x multiplied by cosine of X. Awesome. So it appears for this particular problem we're ultimately trying to figure out and determine the 9-seventh derivative of 3 X multiplied by cosine of X. So now that we know that we're trying to ultimately calculate and determine this 97th derivative of this specific function, what we need to do first off is we need to compute the first few derivatives. So, for our first derivative, so let, we're gonna do perform the derivative a few times. We're gonna do the first derivative. The second derivative. Third derivative. 4th derivative and 5th derivative to help us establish a pattern. So we're gonna take the derivative 5 different times to help us figure out what sort of pattern is taking place. So, or rather to establish a pattern that's taking place. So, for our first derivative, we need to take D by DX of 3 X multiplied by cosine of X. Fantastic, which will be equal to 3 X multiplied by negatives of X. Plus co-sign. Of X multiplied by 3, which will be all equal to when we simplify, -3X multiplied by sin of X. Plus 3 cosine of x. Awesome. So, just to move it over just a little bit, so it's not off screen here. So once again, it's going to be all equal to -3 X multiplied by sin of X plus 3 cosine of X or 3 multiplied by cosine of X. Awesome. So since you should know how to perform basic derivatives, I'm gonna skip the derivation process for the, I'm just gonna skip a few steps, but let us know that for the second derivative, we're gonna be taking. The by. So it's gonna be Dscript 2 by DX Superscript 2, so our second derivative of 3 X multiplied by cosine. of X is going to be equal to, so essentially, when we take the second derivative, we will find that it's gonna be ultimately, since I'm gonna skip a few steps, it's gonna be ultimately equal to in its most simple form, -3. X multiplied by cosine. Of X minus 6 multiplied by sin of X. And essentially what we're doing is when we took our first derivative, the answer that we get for our first derivative, we're going to be taking the derivative of that answer, and when we do, we will get -3 X multiplied by cosine of X minus 6 multiplied by sin of X. And for our third derivative, so Dscript 3 by DX 3. Of 3 X multiplied by cosine of X is going to be ultimately equal to in its most simple form, 3 X multiplied by sin of X minus 9 multiplied by cosine. Of X, and for our 4th derivative, so Dscript 4 by DX 4. We're going to perform that of, so we're taking D to the power of 4 by, you know, per, so as we're taking the 4 derivatives, so D to the superscript of 4 by DX 4 of 3 X multiplied by cosine of X is going to be ultimately equal to in its most simple form, 3 X multiplied by cosine of X plus 12 multiplied by sine of X. And finally, our fifth derivative, so D to the power of 5, or D 5 by DX 5 of 3 X multiplied by cosine of X is going to be ultimately equal to -3X multiplied by sin of X plus 15 multiplied by cosine of X. Fantastic. So now, Our next step is we need to observe our results, so when we observe our derivatives, we will see that the first term in our original function, which is 3 X multiplied by cosine of X, repeats every 4th derivative, and the second term follows this particular pattern. So that will mean. And I write our results in red, so a 0 goes to 3 of 3 multiplied by 1, multiplied by cosine of X goes to -3, multiplied by 2, multiplied by sin of X goes to -3, multiplied by 3, multiplied by cosine of X. Which goes to 3, multiplied by 4, multiplied by sine of X, which goes to 3, multiplied by 5, multiplied by cosine of X. Fantastic. So now the numerical coefficient of the 2nd term when. And mod. 4 is equal to 1. is +3 N and when N mod. 4 is equal to 2. It is -3, and when N mod. is equal to 3, that will mean that it is -3 and And when And So I said, so when and so essentially. I've left out one, so essentially they'll quickly recap. So once again, so when N mod 4 is equal to 1, then it's going to be plus 3N. When N mod 4 is equal to 2, it's going to be equal to -3N. So when N mod 4 is equal to 3, it will be -3N. And finally, when N mod 4 is equal to 0, it will be plus 3N. So, in order to determine the 97th derivative of 3 X multiplied by cosine of X, we need to calculate 97 mod 4, so we need to take 97 mod 4 is equal to 1. So, therefore, the 97th derivative of 3 X multiplied by cosine of X follows the pattern of the first derivative where the coefficient of the second term is plus 3N. So therefore, without a doubt, we could go ahead and write that D 97 per DX 97. Is going to be equal, or so we're gonna be taking the ninety-seventh derivative of. 3 X multiplied by cosine of X. Which is going to be equal to. -3 X multiplied by sign of. X, and it's gonna be plus. So, plus 3 multiplied by 97, so you take 3 multiplied by 97, so 3 multiplied by 97, multiplied by cosine. Of X, which will be equal to when we perform that calculation and simplify, it's gonna be simply equal to -3X multiplied by sin of X plus 291 multiplied by cosine of X, and that will be our final answer. Hooray, we did it. So to quickly recap here, Our final answer, once again, and I'll write our final answer in green, is going to be -3. X multiplied by sin of X plus 291 multiplied by cosine of X, and that is our final answer. Hooray, we did it. So therefore, the 97th derivative of 3 X multiplied by cosine of X will once again be -3X multiplied by sin of x plus 291 multiplied by cosine of X. Thank you so much for watching. Hopefully that helped, and I can't wait to see you in the next video. Bye.

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

c. d⁷³/dx⁷³ (x sin x)

Key Concepts

Product Rule

Higher Order Derivatives

Pattern Recognition in Derivatives

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