From 183c55c09e62c069d65b2f5d9f5d7c065ad16228 Mon Sep 17 00:00:00 2001
From: Christian Kolset <christian.kolset@gmail.com>
Date: Wed, 3 Sep 2025 13:19:49 -0600
Subject: Updated entries

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 tutorials/module_3/numerical_differentiation.md | 67 -------------------------
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-# Numerical Differentiation
-Finding a derivative of tabular data can be done using a finite difference. Here we essentially pick two points on a function or a set of data points and calculate the slope from there. Let's imagine a domain $x$ as a vector such that $\vec{x}$ = $\pmatrix{x_0, x_1, x_2, ...}$. Then we can use the following methods to approximate derivatives
-
-## Forward Difference
-Uses the point at which we want to find the derivative and a point forwards on the line.
-$$
-f'(x_i) = \frac{f(x_{i+1})-f(x_i)}{x_{i+1}-x_i}
-$$
-*Hint: Consider what happens at the last point.*
-
-```python
-import numpy as np
-import matplotlib.pyplot as plt
-
-# Initiate vectors
-x = np.linspace(0, 2, 100) 
-y = 34 * np.exp(3 * x)
-
-dydx =  (y[1:] - y[:-1]) / (x[1:] - x[:-1])
-
-# Plot the function
-plt.plot(x, y, label=r'$y(x)$')
-plt.plot(x, dydx, label=b'$/frac{dy}{dx}$')
-plt.xlabel('x')
-plt.ylabel('y')
-plt.title('Plot of $34e^{3x}$')
-plt.grid(True)
-plt.legend()
-plt.show()
-```
-
-
-## Backwards Difference
-Uses the point at which we want to find
-$$
-f'(x_i) = \frac{f(x_{i})-f(x_{i-1})}{x_i - x_{i-1}}
-$$
-
-
-```python
-import numpy as np
-import matplotlib.pyplot as plt
-
-# Initiate vectors
-x = np.linspace(0, 2, 100) 
-y = 34 * np.exp(3 * x)
-
-dydx =  (y[1:] - y[:-1]) / (x[1:] - x[:-1])
-
-# Plot the function
-plt.plot(x, y, label=r'$y(x)$')
-plt.plot(x, dydx, label=b'$/frac{dy}{dx}$')
-plt.xlabel('x')
-plt.ylabel('y')
-plt.title('Plot of $34e^{3x}$')
-plt.grid(True)
-plt.legend()
-plt.show()
-```
-## Central Difference
-
-$$
-f'(x_i) = \frac{f(x_{i+1})-f(x_{i-1})}{x_{i+1}-x_{i-1}}
-$$
-
-
-
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