Find the n th derivative of: (1)cos4x

Sure, let's find the nth derivative of cos(4x).

First, let's start with the first few derivatives to identify a pattern:

1st derivative: The derivative of cos(4x) is -4sin(4x).

2nd derivative: The derivative of -4sin(4x) is -16cos(4x).

3rd derivative: The derivative of -16cos(4x) is 64sin(4x).

4th derivative: The derivative of 64sin(4x) is 256cos(4x).

From these calculations, we can see a pattern emerging. The nth derivative of cos(4x) will depend on whether n is even or odd, and it will also depend on the remainder when n is divided by 4.

Here's the general formula:

• If n is even, the nth derivative will be 4^n * cos(4x) if n/4 leaves a remainder of 0, and 4^n * -cos(4x) if n/4 leaves a remainder of 2.

• If n is odd, the nth derivative will be 4^n * -sin(4x) if n/4 leaves a remainder of 1, and 4^n * sin(4x) if n/4 leaves a remainder of 3.