nth order derivatives of two functions

To determine the nth order derivative of a rational function, we may have to

i) split it into partial fractions

ii) apply Demoivre's theorem
i.e, (cos θ ± i sin θ)n = cos nθ ± i sin nθ
where 'n' is an integer and i = √–1

Let f and g be two real-valued functions defined as E ⊆ R.
Let both f and g have nth order derivatives over E.
Then we have,

i) for any constant 'k' the nth order derivative of kf exists over E and
(kf)(n) = k f(n)

ii) the nth order derivative of (f + g) exists over E and
(f + g)(n) = f(n) + g(n)

iii) the nth order derivative of (f – g) exists over E and
(f – g)(n) = f(n) – g(n)

iv) the nth order derivative of the product function fg is given by Leibnitz theorem (named after Gottfried Wilhelm Leibnitz, a German mathematician and philosopher of 17th century).