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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
+
+```
+