Exercise. Find the first nine derivatives of the function *f* (*x*) = sin *x*.

Answer

The derivatives of *f* (*x*) = sin *x* follow a pattern. Every 4 derivatives, we find the original function. *f* (*x*) is the same as *f* ^{(4)}(*x*) is the same as *f* ^{8}(*x*).

Similarly, *f* ^{(1)}(*x*) is the same as *f* ^{(5)}(*x*) is the same as *f* ^{9}(*x*) .

If two numbers differ by a multiple of 4, the derivatives of sin *x* corresponding to those numbers will be the same.