Calculate the first five derivatives

Question

Calculate the first five derivatives of f(x)=cosx.

Then determine f^(8)(x) and f^(37)(x)

Answer 1

Function	f(x) = cos(x)	1st Derivative	f '(x) = −sin(x)
1st Derivative	f '(x) = −sin(x)	2nd Derivative	f ''(x) = −cos(x)
2nd Derivative	f ''(x) = −cos(x)	3rd Derivative	f '''(x) = sin(x)
3rd Derivative	f '''(x) = sin(x)	4th Derivative	f ''''(x) = cos(x)
4th Derivative	f ''''(x) = cos(x)	5th Derivative	f 5'(x) = −sin(x)
5th Derivative	f 5'(x) = −sin(x)	6th Derivative	f 6'(x) = −cos(x)
6th Derivative	f 6'(x) = −cos(x)	7th Derivative	f 7'(x) = sin(x)
7th Derivative	f 7'(x) = sin(x)	8th Derivative	f 8'(x) = cos(x)
8th Derivative	f 8'(x) = cos(x)	9th Derivative	f 9'(x) = −sin(x)

Every 4th derivative, it repeats. Notice the 8th derivative is also cos(x). So will the 12th, 16th, . . . 32nd, and 36th. So if the 36th derivative is cos(x), then the 37th will be -sin(x)