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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
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