Question

How do you compute the 200th derivative of `f(x)=sin(2x)`?

Answer 1

`2^200 sin(2x)`

Explanation:

`f(x) = sin(2x) implies`
`d/dx f(x) = 2 cos(2x) implies`
`d^2/dx^2 f(x) = -2^2 sin(2x) implies`
`d^3/dx^3 f(x) = -2^3 cos(2x) implies`
`d^4/dx^4 f(x) = 2^4 sin(2x) implies`

This means that differentiating `f(x)` 4 times results in the same function, with a multiplying factor of 4.

Differentiating 200 times is the same as repeating the above 50 times, and so

`d^200/dx^200 f(x) = (2^4 )^50sin(2x) = 2^200 sin(2x)`