Question

How do you find the 27th derivative of cosx?

Answer 1

`(d^27y)/(dx^27) = sinx`

Explanation:

Consider the following pattern.

`d/dx(cosx) = -sinx`

`d/dx(-sinx) = -cosx`

`d/dx(-cosx) = sinx`

`d/dx(sinx) = cosx`

So, after differentiating `cosx` four times, you will return to `cosx`!!

Make a list:

`0. cosx`
`1. -sinx`
`2. -cosx`
`3. sinx`
`4. cosx`

So, whenever you have an nth derivative where `n` is divisible by `4`, the derivative will be equal to `cosx`. The closest multiple of `4` to `27` is `28`. The 28th derivative of `cosx` is `cosx`. Go one up in the list `(3)` to find that the 27th derivative of `cosx` is `sinx`.

Hopefully this helps!