We’ve discussed how to compute derivatives by hand. However, oftentimes when working on a computer, we have functions given to us in array form rather than an explicit equation. In this exercise, we will investigate how to calculate derivatives using Python.

For example, let’s say we have an array that represents the following function:

$f(x) = x^2 + 3$

We can define this function in two sets of arrays using the following code.

from math import pow
# dx is the "step" between each x value
dx = 0.05
def f(x):
  # to calculate the y values of the function
  return pow(x, 2) + 3
# x values
f_array_x = [x for x in np.arange(0,4,dx)]
# y values
f_array_y = [f(x) for x in np.arange(0,4,dx)]

The pow() function within the f(x) function we defined allows us to calculate the values of x², and we use a list comprehension to create a list of the y-values in a variable called f_array_y.

To compute the derivative of f_array, we use a NumPy function called gradient().

f_array_deriv = np.gradient(f_array_y, dx)

gradient() takes in an array (in this case, a one-dimensional array) as its first argument. We also need to specify the distance between the x values, which is dx. For a one-dimensional array, gradient() calculates the rate of change using the following formula:

$\frac{\Delta y}{\Delta x} = \frac{Change\ in\ y}{Change\ in\ x}$

As we know from the limit definition of the derivative, we want “Change in x” to be as small as possible, so we get an accurate instantaneous rate of change for each point.