Find the 119th derivative for f(x) = cos(x) and f(x) = sin(x) respectively.

sin(x)

- cos(x)

Step-by-step explanation:

The derivative for f(x) up to 4 terms is

f(x) = cos(x)

f'(x) = - sin(x)

f''(x) = - cos(x)

f'''(x) = -(-sin(x))

f'''(x) = sin(x)

f''''(x) = cos(x)

What this is tell you is that you need to go to 4 differentiations before you get back to where you started from.

The next step is to find out how many groups of 4 there are in 119 differentiations, and, more importantly, what the remainder is.

So you have to go through 29 differentiations to get to 116 times you have differentiated (119/4 = 29)

There are 3 more differentiations you have to do which will be f'''(x) = sin(x)

The answer is

===============

f(x) = sin(x)

f'(x) = cos(x)

f''(x) = -sin(x)

f'''(x) = -cos(x)

f''''(x) = sin(x)

The argument is going to be the same as you used above. 116 differentiations will get you back to sin(x). You need 3 more differentiations so f'''(x) = - cos(x)