Polynomial elements Variable X at which we find values of a derivative
 Given function

Consider one of the simple and undeservedly forgotten on the Internet Internet methods for determining the derivative of a polynomial, an arbitrary (positive) degree.

Until recently, I was sure that if a polynomial of the form

$f(x)=a_0x^{n}+a_1x^{n-1}+a_2x^{n-2}+.....+a_{n-1}x+a_n$

and it is necessary to find out the value of a derivative, for example, of the 5th order at some point, you must first calculate this derivative (of the fifth order), and then substitute the value, calculate the derivative.

It turns out there is a simpler and algorithmically easier way to find the derivative at a point.

To do this, we need the technique described in the materials: Expand the polynomial in degrees and Horner method online calculator. Division of a polynomial.

Yes, yes, it turns out the Horner method successfully solves the problem.

Consider an example:

Calculate the third-order derivative for x = 3 of the next polynomial

$f(x)=2x^3-5x^2+x+7$

1. Divide the given polynomial by $x-3$

Get $f(x)=2x^2+x+4$ and the remainder of 19.

The number 19 is the value of the function $f(x)=2x^3-5x^2+x+7$ if we substitute x = 3 there

2. Divide $f(x)=2x^2+x+4$ again on $x-3$

Get $f(x)=2x+7$ and the remainder of 25.

Since this is the first result, we multiply the result by 1! (One factorial) = 1. Got the same number 25

The number 25 is the value of the first derivative of the given function for x = 3. That is, if we calculate the first derivative

$f'(x)=6x^2-10x+1$ and substitute the value 3 there, we get the same answer = 25.

3. Divide $f(x)=2x+7$ again on $x-3$

we get $f(x)=2$ and residue 13.

Multiply this number by 2! (two factorial) = 2 and we get the value of the derivative of the second order function for x = 3

This number = 26

4. The third-order derivative is calculated in this case simply, since $f(x)=2$ it’s impossible to divide further, this is the remainder. It must be multiplied by 3! (Three factorial) = 6

And we get that the third-order derivative for a given polynomial for x = 3 is 12.

In such a straightforward way, we can find the values ​​of any derivative of any polynomial.

The algorithm is simple, but with polynomials with degrees above 10, we are faced with the need to calculate factorials above 10, which is very laborious, since the factorial from 10 is 3628800 and the factorial from 16 is already 20922789888000

But we benefit from one of the properties of Horner's methodology, which states: If we multiply a function by a number, then the remainder of the branch will increase by the same amount.

Therefore, it is enough for us to multiply the obtained coefficients of the polynomial by dividing by the numbers 1,2,3,4,5, etc. depending on which derivative we’ll calculate at the moment and calculate the remainder.

The calculator also works in the field of complex numbers, so let's solve this example.

There is a function $f(x)=2x^7+(1-5i)x^6 -7x^4+x^3i+2x^2 -9x-1$

It is necessary to find out all possible derivatives of this function for x = i

It is easy to make sure that solving it manually, you can make a mistake and go the wrong way.

It is much easier to use the bot and write through XMPP client

propol 2 1-5i 0 -7 i 2 -9 -1; i

and we will get all the results

The polynomial derivative values ​​are found
0 derivative. Function Value -10-6i
1 derivative. Function Value 7 + 35i
2 derivative. Function Value 112-66i
3 derivative. Function Value -180-282i
4 derivative. Function Value -528 + 120i
5 derivative. Function Value -1440 + 720i
6 derivative. Function Value 720 + 6480i
7 derivative. Function Value 10080

The logical question is what is the zero derivative?
Answer - this is the original function. And the value -10-6i is obtained if we substitute -i into the original function

Let's try to solve another equation
we know what the fourth derivative of the function is equal to $f(x)=(7-i)x^{17}+2x^{11} -ix^7+9x-5$
for x = 2 + i

A polynomial of the 17th degree .. this is serious as well as computation with a complex argument.
Well try
Preset function
$f(x)=(7-i)*x^{17}+(2)*x^{11}+(-i)*x^{7}+(9)*x^{1}+(-5)*x^{0}$
 Derivative The value of the derivative at X = 2 + i 0 707043 + 6123674i one 25630678 + 39273242i 2 289802562 + 169486216i 3 2247959580 + 147950190i 4 13006113720-5465417040i 5 53432793120-62240220840i 6 107126132400-427018989600i 7 -468058852800-2114656795440i 8 -6101588908800-7522728998400i nine -35506871769600-16099283692800i 10 -1.393813225728E + 14 + 5293047513600i eleven -3.828579156864E + 14 + 2.0995438464E + 14i 12 -6.6691392768E + 14 + 9.6332011776E + 14i thirteen -3.705077376E + 14 + 6.1024803840002E + 14i 14 1.4820309504E + 15 + 7.8460462080004E + 14i fifteen 5.2306974720004E + 14 + 5.230697472E + 14i sixteen 3.1384184832005E + 14 + 1.0461394944E + 14i 17 24.89811996672https://abak.pozitiv-r.ru

when x = 2 + i, the value of the function when taking the fourth derivative will be
 4 13006113720-5465417040i
What else can you notice?
What you need to carefully look at the calculations.
In our example, when taking 17 winding, the number 24.898 is obtained
although it should certainly be $(7-i)(17!)$ where is 17! this is the factorial from 17 = 355687428096000
This small flaw (an error in calculating large derivatives) will be eliminated soon. But the calculation of derivatives is not higher than 10 orders of magnitude, the bot performs correctly.

Good luck!

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