(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Find the 73rd derivative of f(x) = sin(2x) + 3

(Hint: Take the first five derivatives to find a pattern)

2. Relevant equations

dy/dx

3. The attempt at a solution

I took the first five derivatives to find the pattern:

dy/dx = 2cos2x

d^{2}y/dx^{2}= -4sin2x

d^{3}y/dx^{3}= -8cos2x

d^{4}y/dx^{4}= 16sin2x

d^{5}y/dx^{5}= 32cos2x

Now if we look at the pattern we can see that every odd term derivative will result in tcos2x and every even term derivative will result in tsin2x where t is an element of all integers.

If we just look at the value of t and not the sign, let's say we are focusing on |t|, then we just look at the pattern and figure out that it will be 2^73 because each of the terms are multiplied by 2^n where n in this case is 73. Now if we were looking at the sign we notice that it alternates: (every underlined term number will be a negative derivative, otherwise it will be a postive derivative)

1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10

11 , 12 , 13 , 14 , 15 , 16 , 17 , 18 , 19 , 20

21, 22 , 23

31, 32 , 33

41, 42 , 43

51, 52 , 53

61, 62 , 63

71, 72 , 73

A pattern starts after the 20th term because then the cycle repeats and by this we know that the 73rd derivative will be a positive derivative so a positive t, thus the 73rd derivative is 2^73cos2x.

I am fairly sure that my answer is correct, however i do not think that my process and explanation are sufficient enough mathematically to show my answer. Can you please help in finding a general formula for this or perhaps a more mathematical way of finding the derivative, especially for the part where i was determining the sign (positive,negative) of the derivative. Thank you.

1. The problem statement, all variables and given/known data

2. Relevant equations

3. The attempt at a solution

1. The problem statement, all variables and given/known data

2. Relevant equations

3. The attempt at a solution

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# Homework Help: Finding a higher order derivative of a trig function

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