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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)=x2+3f(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:

ΔyΔx=Change in yChange in x\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.

Instructions

1.

In script.py, we have code that defines an plots the following function:

f(x)=sin(x)f(x) = sin(x)

sin_x contains the x-values, while sin_y contains the y-values.

Define a variable sin_deriv that calculates the derivative of f(x) = sin(x).

2.

Uncomment the following line of code to plot sin_deriv:

#plt.plot(sin_x, sin_deriv)

What does the new plot look like?

3.

Let’s play around with the dx variable. Change the value of dx from 0.01 to 1.

What do you notice about the graphs after you hit run? Does the accuracy of your calculation appear to increase or decrease?

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