| Polynomial elements |
| Variable X at which we find values of a derivative |
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| 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
=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
1. Divide the given polynomial by
Get =2x^2+x+4)
The number 19 is the value of the function =2x^3-5x^2+x+7)
2. Divide
Get =2x+7)
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
=6x^2-10x+1)
3. Divide
we get =2)
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
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
=2x^7+(1-5i)x^6%20-7x^4+x^3i+2x^2%20-9x-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
=(7-i)x^{17}+2x^{11}%20-ix^7+9x-5)
| Preset function | ||||||||||||||||||||||||||||||||||||||
=(7-i)*x^{17}+(2)*x^{11}+(-i)*x^{7}+(9)*x^{1}+(-5)*x^{0}) |
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| 4 | 13006113720-5465417040i |
(17!))
where is 17! this is the factorial from 17 = 355687428096000