A **derivative** describes the rate of change of functions.  In this answer, we will calculate an equation's Nth-order derivative. We will be using the following equation to calculate the derivative:

$$
f(x) = (1/e^{x}) + (sin^{2}(\frac{x}{2}))
$$



# Steps:
To calculate the Nth-order derivative of the equation, follow the steps given below:
- Import the following modules:
  - `sympy`: This module is used to calculate the derivative of the equation.
  - `numpy`: This module is used to perform the numeric operations on the equation.
  - `matplotlib.pyplot`: This module is used to plot the equation results.

- Define a symbol using the `symbols()` method available in `sympy`.

- Define the equation using the Python expressions.

- Use a loop to iterate for the `N` times and use the `diff()` method from `sympy` to calculate the derivative of the equation.

- Define an array containing some range of values for the x-axis in the plot.

- Use the `subs()` method from the equation to get the results of each substitution of the value in the equation.

- Plot the resultant values with the input values.

# Example code

# import library
import sympy as sp
import matplotlib.pyplot as plt
import numpy as np

# Define the symbols
x = sp.symbols('x')
# Define the equation
equation = (1/sp.exp(x)) + (sp.sin(x/2) ** 2)
print("Equation: ", equation)

# Calculate the derivative
N = 4
for i in range(N):
  equation = equation.diff(x)
  print('Order ' ,i+1 , ": ", equation)

# Plotting the results
time = np.arange(0, 10, 0.1)
solution = [equation.subs({x: point}) for point in time]

# plot derivative time history
plt.plot(time, solution)

How to calculate the Nth-order derivative using Python

Calculate any equation’s Nth-order derivative in Python using sympy, numpy, and matplotlib.