Partial Derivatives
In AI, we often deal with functions that depend on many variables – not just x, but also y, z, and more. For example, the error of a neural network depends on thousands of weights. A partial derivative measures how the function changes when you vary only one variable, keeping all others fixed.
A partial derivative tells you the sensitivity of a multivariable function to changes in a single variable.
Analogy: Baking a Cake
Imagine a cake recipe with many ingredients: flour, sugar, eggs, butter. The cake’s quality depends on all of them. A partial derivative for flour would answer: "If I increase flour by a tiny bit (keeping everything else the same), how much does the cake quality change?"
Why Partial Derivatives Matter in AI
- Neural network training: The loss function depends on every weight. We compute the partial derivative of the loss with respect to each weight.
- Gradient: The vector of all partial derivatives is the gradient (next chapter).
- Multivariate optimization: Finding the minimum of a function with many inputs.
Notation
If f(x, y) = x² + 3xy, then:
- Partial derivative with respect to x (treat y as constant): ∂f/∂x = 2x + 3y.
- Partial derivative with respect to y (treat x as constant): ∂f/∂y = 3x.
Visualizing Partial Derivatives
Think of a hilly landscape (altitude depends on x and y coordinates). The partial derivative with respect to x tells you how steep the hill is if you walk east‑west. The partial derivative with respect to y tells you steepness if you walk north‑south.
Example: Simple Neural Network
A neuron computes output = activation(w₁·x₁ + w₂·x₂ + b). The loss depends on w₁, w₂, b. To update w₁, we need ∂Loss/∂w₁ – a partial derivative.
Two Minute Drill
- Partial derivative = derivative with respect to one variable, holding others constant.
- Used to compute how each weight affects the loss.
- Notation: ∂f/∂x, ∂f/∂y.
- The vector of all partial derivatives is the gradient.
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