m-th derivative polynomial zero

$m$-th derivative of a polynomial function at $x=0$

Using limit n-th derivative of a polynomial function can be easily found ¹. Some theorems, such as power rule, sum & different rule, and constant multiple rule, are required in finding the derivatives ². Using derivative critical and inflection points of a polynomial function can be studied ³.

Here a formula for $m$-th derivatives at $x=0$ of a polynomial function and is given.

A polynomial function can be presented in a form of

$$\tag{1} f(x) = \sum_{i=0}^n a_i x^i. $$

First derivative of $f(x)$ is simply

$$\tag{2} f'(x) = \sum_{i=1}^n i \ a_i \ x^{i-1}, $$

second derivative is

$$\tag{3} f''(x) = \sum_{i=2}^n (i-1) i \ a_i \ x^{i-2}, $$

third derivative is

$$\tag{4} f'''(x) = \sum_{i=3}^n (i-2)(i-1) i \ a_i \ x^{i-3}, $$

and fourth derivative is

$$\tag{5} f'^\nu(x) = \sum_{i=4}^n (i-3)(i-2)(i-1) i \ a_i \ x^{i-4}, $$

which can be later generalized until $m$-th derivative as

$$\tag{6} f^m(x) = \sum_{i=m}^n (i-m+1) \cdots (i-3)(i-2)(i-1) i \ a_i \ x^{i-m}. $$

For $x=0$ it can be obtained that

$$\tag{7} f^m(0) = m! \ a_m, $$

since the other terms are zero due to existence of $x^i$ with $i > 0$.