Question

Let `f(x) = sin(x)`. Find `f^400(x)` (the `400th` derivative of `f(x)`)?

Answer 1

`sin x`

Explanation:

We know that `e^(ix)=cosx+isinx` and

`sinx = "Im"(e^(ix))` now

`(d^n)/(dx^n) e^(ix)=i^n e^(ix)`

now if `n=400` we have `i^400=(i^4)^100=1`

so finally

`f^((400))(x)=sin x`

Answer 2

`f^400 (x) = sinx`

Explanation:

Here's another way.

We create a table with the n derivatives of `f(x)`.

`f(x) = sinx`
`f^1(x) = cosx`
`f^2(x) = -sinx`
`f^3(x) = -cosx`
`f^4(x) = sinx`

So, essentially, whenever you have a multiple of `4`, such as `400`, your derivative will be `sinx`.

Therefore, `f^400 (x) = sinx`.

Hopefully this helps!