Derivatives and the Geometry of Change

What You'll Cover Today

Day 1 of Calculus for AI in 5 Days lays the foundation. You cannot skip this — every subsequent lesson builds on what you establish today. Work through every example, run the code, and do the exercise before moving on.

power rule

Understanding power rule is the core goal of Day 1. 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.

# power rule — Working Example
# Study this pattern carefully before writing your own version

class powerruleExample:
    """
    Demonstrates core power rule 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': 'power rule',
            'data': self.config
        }
        return result


# Usage
example = powerruleExample({
    'name': 'my-implementation',
    'type': 'power rule'
})
output = example.process()
print(output)

product rule

product rule is the practical application of power rule in real projects. Once you understand the underlying model, product rule becomes the natural next step.

limits

limits rounds out today's lesson. It connects power rule and product rule into a complete picture. You'll use all three concepts together in the exercise below.

Common Mistakes on Day 1

• Skipping the fundamentals — power rule requires understanding the underlying model before you can apply it correctly. Read the section twice if needed.
• Ignoring error messages — error messages for product rule 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?