How do you find the 50th derivative of #y=cos2x#?

1 Answer
Jake M.
Mar 7, 2018

#y^((50)) = -2^(50) cos(2x)#

Explanation:

A fiftieth derivative! Guess we better start going...

Zeroth derivative: #y = cos(2x) #
First derivative: #y' = -2sin(2x) #
Second derivative #y'' = -4 cos(2x) #

Wait... this second derivative is familiar. It's -4 times the original function! That means that if we differentiation two more times, we will just be multiplying by -4, i.e. we already know that
#y^((4)) = 16 cos(2x)#

That pattern makes this a lot easier! Every two derivatives give us a -4, so that means
#y^((2n)) = (-4)^n cos(2x) #

At #n=25#,
#y^((50)) = (-4)^25 cos(2x) #
Since 25 is odd, this is negative. Using #4 = 2^2#, this shows us the final answer
#y^((50)) = -2^(50) cos(2x), #
our final answer.

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