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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. You may have done this before in spreadsheets, we're going to do this using python. Let's imagine a time range $t$ as a vector such that $\vec{t}$ = $\pmatrix{t_0, t_1, t_2, ...}$ and a displacement domain as a function of time. We can represent the range and domain as two python arrays `t` and `s` respectively.
-
-```python
-import numpy as np
-
-# Initiate time domain
-t = np.linspace(0, 2, 100)
-s = 34 * np.exp(3 * t)
-```
-
-Then we can use the following methods to approximate the definitive derivatives as follows.
-
-## Finite Difference
-### Forward Difference
-Uses the point at which we want to find the derivative and a point forwards in the array.
-$$
-f'(x_i) = \frac{f(x_{i+1})-f(x_i)}{x_{i+1}-x_i}
-$$
-*Note: If we apply this to an array, consider what happens at the last point.*
-
-```python
-for s in t:
-	ds_dt = f(x+)
-```
-
-
-
-```python
-# Forward difference using python arrays
-dsdt =  (s[1:] - s[:-1]) / (t[1:] - t[:-1])
-
-import matplotlib.pyplot as plt
-
-# Plot the function
-plt.plot(t, s, label=r'$s(t)$')
-plt.plot(t, dsdt, label=b'$/frac{ds}{dt}$')
-plt.xlabel('Time (t)')
-plt.ylabel('Displacement (s)')
-plt.title('Plot of $34e^{3t}$')
-plt.grid(True)
-plt.legend()
-plt.show()
-```
-
-
-### Backwards Difference
-Uses the point at which we want to find the derivative and the previous point in the array.
-$$
-f'(x_i) = \frac{f(x_{i})-f(x_{i-1})}{x_i - x_{i-1}}
-$$
-
-
-```python
-dsdt =  (s[1:] - s[:-1]) / (t[1:] - t[:-1])
-
-# Plot the function
-plt.plot(t, s, label=r'$s(t)$')
-plt.plot(t, dydx, label=b'$/frac{ds}{dt}$')
-
-plt.xlabel('Time (t)')
-plt.ylabel('Displacement (s)')
-plt.title('Plot of $34e^{3t}$')
-plt.grid(True)
-plt.legend()
-plt.show()
-```
-Try plotting both forward and backwards
-
-### Central Difference
-$$
-f'(x_i) = \frac{f(x_{i+1})-f(x_{i-1})}{x_{i+1}-x_{i-1}}
-$$
-### Problem 1
-Use the forward difference formula to approximate the derivative of $f(x)$ at $x = 1$ using step sizes: $h=0.5$ and $h=0.1$ for the following function.
-$$
-f(x) = \ln(x^2 + 1)
-$$
-Compare your results with the analytical solution at $x=1$. Comment on how the choice of $h$ affects the accuracy.
-```python
-
-```
-### Problem 2
-Use the central difference formula to approximate the derivative of $f(x)$ at $x = 1.2$ using step sizes: $h=0.5$ and $h=0.1$ for the following function.
-$$
-f(x) = e^{-x^2}
-$$
-Compare your results with the analytical solution at $x=1.2$. Comment on how the choice of $h$ affects the accuracy.
-```python
-
-```
-
-
----
-# Advanced Derivatives
-
-## High-Accuracy Differentiation Formulas
-## Richardson Extrapolation
-## Derivative of Unequally Spaced Data 
-## Partial Derivatives
-