Derivatives In PYTHON (Symbolic AND Numeric)

Summary

In this video, the process of taking derivatives in Python is explored through various scenarios: symbolic derivatives, numerical derivatives from data, and quasi-symbolic derivatives. The video demonstrates how Python packages like SymPy, NumPy, and SciPy can simplify these tasks, with a focus on real-world applications like noisy data and field data collection. It also covers techniques like data smoothing and applying filters to improve the accuracy of derivative calculations. The tutorial emphasizes the power of Python for handling complex derivative computations efficiently, making it a valuable tool for researchers and students alike.

Takeaways

• 😀 Symbolic derivatives are computed when you have a function and a clear formula, making it easy to differentiate using Python libraries like SymPy.
• 😀 For symbolic differentiation, libraries like SymPy allow for handling complex functions that would be tedious to differentiate manually, such as e^(-a sin(x)^2) * sin(b^x) * log(c * sin(x)^2 / x).
• 😀 The derivative of a symbolic expression can be computed in just one line of code using SymPy's diff function, which can also compute higher-order derivatives.
• 😀 You can evaluate derivatives at specific values by substituting constants using the substitute method and obtaining numerical results with evalf.
• 😀 For data-based derivatives (e.g., noisy real-world data), Numpy’s gradient function can be used to approximate the derivative from discrete data points.
• 😀 It's essential to smooth noisy data before computing derivatives, as noise in data can lead to highly erratic and misleading derivative plots.
• 😀 The derivative of noisy data can be computed after smoothing the data using techniques like convolution with a moving average window to reduce noise amplification.
• 😀 When dealing with data collected from experiments, it’s useful to visualize the derivative alongside the original data to interpret trends and rate of change.
• 😀 In the quasi-symbolic case, when you don’t have a symbolic formula but a functional relationship (e.g., a maximum distance between a curve and data), the derivative can be numerically approximated using libraries like SciPy.
• 😀 Python’s np.vectorize function can be used to apply derivative functions over an array of values, automating the process of calculating derivatives for complex or non-analytic functions.

Q & A

What are the three main cases for taking derivatives in Python?

The three main cases are: 1) Symbolic derivatives, where you have a clear mathematical function and can apply calculus rules. 2) Numerical derivatives, which involve data collected in the field, where you calculate the derivative of noisy or discrete data. 3) Quasi-symbolic derivatives, which occur when the function is complex and not in a simple mathematical form, but you can still compute the derivative numerically.

What package is primarily used for symbolic derivatives in Python?

The primary package used for symbolic derivatives is SymPy (referred to as 'simp' in the video). SymPy allows you to define symbolic variables and perform calculus operations, such as differentiation, easily in Python.

How can you take the derivative of a function symbolically in Python?

You can take the derivative symbolically by using SymPy’s `diff` function. First, define the symbols with `symbols()`, then define your function. Finally, use `diff()` on the function to compute its derivative with respect to a variable.

How do you evaluate the derivative at specific values in SymPy?

After computing the symbolic derivative, you can use the `subs()` function to substitute specific values for the symbols and then use `evalf()` to evaluate the result as a float, which gives you the numerical value of the derivative at those points.

What is the difference between symbolic and numerical derivatives?

Symbolic derivatives involve exact mathematical expressions and are calculated using algebraic rules. Numerical derivatives, on the other hand, are computed based on discrete data, typically involving algorithms like finite differences or gradient estimation.

How do you compute numerical derivatives of data in Python?

For numerical data, you can use NumPy's `gradient()` function to compute the derivative. This function calculates the gradient of a dataset, representing the rate of change of the values with respect to the independent variable.

What is the issue when computing derivatives of noisy data?

When you compute the derivative of noisy data, the noise is amplified, resulting in a noisy derivative. This can make it hard to interpret the true trend or rate of change of the data.

How can you reduce noise when calculating the derivative of noisy data?

You can smooth the data before calculating the derivative. One common method is using a moving average or convolution filter, such as applying a rectangular kernel to average the data points within a window. This helps to reduce the noise in the dataset and yields a smoother derivative.

What is the 'quasi-symbolic' case mentioned in the video?

The quasi-symbolic case refers to situations where you know the general functional form of a relationship (like a complex equation or numerical function) but cannot compute the derivative using traditional symbolic methods. Instead, you can use numerical techniques to approximate the derivative.

How do you compute derivatives for quasi-symbolic functions in Python?

For quasi-symbolic functions, you can evaluate the function at different points and use numerical methods to compute the derivative. You can apply techniques like `np.gradient()` for simple cases or use more advanced methods such as SciPy's `derivative()` function for better precision.