- #1

tmlrlz

- 29

- 0

## Homework Statement

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

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

## Homework Equations

dy/dx

## 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.