what is the 50th derivative of f(x) = cos x ?

Question

please show work if possible...

Answer 1

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

this pattern repeats.

the fiftieth deirvative is - cos x

Answer 2

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

The cycle starts over again, so the fifth derivative is -sin(x), the sixth is -cos(x), and so forth. Every multiple of 4'th derivative is cos(x).

... So the 48th derivative is cos(x)
That means the 49th derivative would be -sin(x)
So the 50th derivative is -cos(x).

Answer 3

cos x goes to -sin x which goes to -cos x which goes to sin x which goes to cos x... so it's follows a pattern and comes back to itself every 4th derivation. using this, 50/4 is 12.5. so basically you're going around in 12 circles and then have 2 more derivations:

cos x --> -sin x... then -sin x --> -cos x and that's your answer.