Question

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

Answer 1

`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.