Backpropagation from First Principles

What You'll Cover Today

Day 3 of Calculus for AI in 5 Days is the midpoint — and often the most rewarding day. The pieces from Day 1 and Day 2 start connecting. Most students have an 'it clicks' moment on Day 3.

partial derivatives

Understanding partial derivatives is the core goal of Day 3. The concept is straightforward once you see it in practice — most confusion comes from skipping the mental model and jumping straight to implementation. Start with the model, then write the code.

# partial derivatives — Working Example
# Study this pattern carefully before writing your own version

class partialderivativesExample:
    """
    Demonstrates core partial derivatives concepts.
    Replace placeholder values with your real implementation.
    """
    
    def __init__(self, config: dict):
        self.config = config
        self._validate()
    
    def _validate(self):
        required = ['name', 'type']
        for field in required:
            if field not in self.config:
                raise ValueError(f"Missing required field: {field}")
    
    def process(self) -> dict:
        # Core logic goes here
        result = {
            'status': 'success',
            'topic': 'partial derivatives',
            'data': self.config
        }
        return result


# Usage
example = partialderivativesExample({
    'name': 'my-implementation',
    'type': 'partial derivatives'
})
output = example.process()
print(output)

gradient

gradient is the practical application of partial derivatives in real projects. Once you understand the underlying model, gradient becomes the natural next step.

directional

directional rounds out today's lesson. It connects partial derivatives and gradient into a complete picture. You'll use all three concepts together in the exercise below.

Common Mistakes on Day 3

• Skipping the fundamentals — partial derivatives requires understanding the underlying model before you can apply it correctly. Read the section twice if needed.
• Ignoring error messages — error messages for gradient are usually precise. Read them carefully before searching online.
• Hard-coding values — anything that might change between environments belongs in configuration, not in your source code.
• Not testing edge cases — the happy path is not enough. What happens with empty input? With unexpected types? With network failures?