Answer:
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 [tex]f^{119} = sin(x)[/tex]
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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)
