Calculating Derivatives of a Function in Python

To compute the first, second, and third derivatives of a function in Python, you can use the diff() function from the SymPy library, which is designed for symbolic mathematics.

diff(f, x, n)

Here, f is the function to differentiate, x is the variable, and n is the order of the derivative.

Below is a complete example of Python code to calculate the first, second, and third derivatives of a function.

First, import the SymPy library:

import sympy as sp

Next, define the symbol for the variable you want to differentiate:

x = sp.symbols('x')

Then, define the function:

f = x**4 + 3*x**3 + 2*x**2 + x + 1

Now, you can proceed with calculating the derivatives.

Use sp.diff(f, x) to calculate the first derivative of the function:

f_prime = sp.diff(f, x)

By default, sp.diff computes the first derivative if the order is not specified.

The result is the first derivative of f(x):

print(f_prime)

4*x**3 + 9*x**2 + 4*x + 1

To calculate the second derivative, differentiate the first derivative:

The second derivative, f''(x), is the derivative of the first derivative, f'(x):

f_second = sp.diff(f_prime, x)
print(f_second)

12*x**2 + 18*x + 4

Alternatively, you can calculate the second derivative directly from the original function by specifying the order as 2, which is often the preferred method:

The result is the same:

f_second = sp.diff(f, x, 2)
print(f_second)

12*x**2 + 18*x + 4

Similarly, you can compute the third derivative:

f_third = sp.diff(f, x, 3)
print(f_third)

24*x + 18

You have now calculated the first, second, and third derivatives of the function f(x).

The same process can be used to find the fourth, fifth, or any higher-order derivative of any function.

And so on.