Find the 73rd derivative of f(x) = sin(2x) + 3
(Hint: Take the first five derivatives to find a pattern)
The Attempt at a Solution
I took the first five derivatives to find the pattern:
dy/dx = 2cos2x
d2y/dx2 = -4sin2x
d3y/dx3 = -8cos2x
d4y/dx4 = 16sin2x
d5y/dx5 = 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.