Find the 𝑘𝑡ℎ derivative of 𝑓(𝑥) = 7𝑥6 − 6𝑥5 − 𝑥4 + 2𝑥3 + 3𝑥 − 2 for every positive integer 𝑘. Use this to generalize the higher order derivatives of an arbitrary polynomial function.

Answer 1

Just to make sure, let me state the properties of the factorial function used below:

0! = 1

1! = 1

2! = 1×2

3! = 1×2×3

etc.

First consider a simple polynomial function

f(x) = xⁱ for some integer i.

Then

f'(x) = ix^i-1

f"(x) = i(i-1)x^i-2

...

f^(k) = i!/(i-k)! x^i-k

This formula is true for k ≤ i, while for k > i, f^(k) = 0

If you are supposed to be formal, you can prove this by induction.

Now apply this to an arbitrary polynomial

F(x) = ∑a_i xⁱ, where i = 0 to n

F^(k) = ∑a_i i!/(i-k)! x^i-k, where i = k to n

This formula is valid even case k > n, because the summation is empty, and F^(k) = 0.