Differentiate a polynomial in Python

To differentiate a polynomial, use the polynomial.polyder() method in Python NumPy. This method returns the polynomial coefficients differentiated m times along a specified axis. The coefficients are ordered from low to high degree, so [1, 2, 3] represents 1 + 2*x + 3*x².

Syntax

numpy.polynomial.polynomial.polyder(c, m=1, scl=1, axis=0)

Parameters

The method accepts the following parameters:

• c − Array of polynomial coefficients from low to high degree
• m − Number of derivatives to take (default: 1, must be non-negative)
• scl − Scaling factor applied at each differentiation (default: 1)
• axis − Axis over which the derivative is taken (default: 0)

Basic Example

Let's differentiate the polynomial 1 + 2x + 3x² + 4x³ ?

import numpy as np
from numpy.polynomial import polynomial as P

# Coefficients for 1 + 2x + 3x² + 4x³
c = np.array([1, 2, 3, 4])

print("Original coefficients:", c)
print("Polynomial: 1 + 2x + 3x² + 4x³")

# First derivative
result = P.polyder(c)
print("First derivative coefficients:", result)
print("Derivative: 2 + 6x + 12x²")

Original coefficients: [1 2 3 4]
Polynomial: 1 + 2x + 3x² + 4x³
First derivative coefficients: [ 2.  6. 12.]
Derivative: 2 + 6x + 12x²

Multiple Derivatives

You can compute higher-order derivatives by specifying the m parameter ?

import numpy as np
from numpy.polynomial import polynomial as P

c = np.array([1, 2, 3, 4])  # 1 + 2x + 3x² + 4x³

# First derivative
first = P.polyder(c, m=1)
print("First derivative:", first)

# Second derivative
second = P.polyder(c, m=2)
print("Second derivative:", second)

# Third derivative
third = P.polyder(c, m=3)
print("Third derivative:", third)

First derivative: [ 2.  6. 12.]
Second derivative: [ 6. 24.]
Third derivative: [24.]

Using Scaling Factor

The scaling factor is useful for linear changes of variables ?

import numpy as np
from numpy.polynomial import polynomial as P

c = np.array([1, 2, 3])  # 1 + 2x + 3x²

# Normal derivative
normal = P.polyder(c)
print("Normal derivative:", normal)

# With scaling factor of 2
scaled = P.polyder(c, scl=2)
print("Scaled derivative (scl=2):", scaled)

Normal derivative: [2. 6.]
Scaled derivative (scl=2): [ 4. 12.]

How It Works

For a polynomial with coefficients [a?, a?, a?, ..., a?] representing a? + a?x + a?x² + ... + a?x?, the derivative has coefficients [a?, 2a?, 3a?, ..., na?] representing a? + 2a?x + 3a?x² + ... + na?x^(n-1).

Conclusion

Use numpy.polynomial.polynomial.polyder() to differentiate polynomials efficiently. The method supports multiple derivatives, scaling factors, and works with multi-dimensional coefficient arrays for complex polynomial operations.