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

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/ex)+(sin2(x2))f(x) = (1/e^{x}) + (sin^{2}(\frac{x}{2})) f(x)=(1/ex)+(sin2(2x​)) 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 