🏔️ You've climbed the whole Differentiation & Integration mountain. Here's everything in one view.

The two halves of calculus:
• Differentiation = slope = rate of change. Sign tells direction; zero marks peaks and valleys.
• Integration = area under the curve = adding things up. The reverse of differentiation.

The rules for slopes:
• Power / exp / sin — the short list worth recognizing.
• Product rule for two functions multiplied (the PSP alpha function).
• Chain rule for functions nested inside functions (and for how one variable changes with another through time).

When there's no clean formula — go numerical:
• Derivative ≈ finite difference, np.gradient.
• Integral ≈ Riemann sum, np.cumsum(y * dt).
• Smaller step = more accurate but more computation. Always a tradeoff.

Two variables or more:
• Partial derivatives (∂) — wiggle one input, freeze the rest. Stack them into the gradient, the heart of model training.

Two ways to picture each operation:
• Geometry: derivative = slope, integral = area.
• Filtering: differentiation = high-pass (keeps fast changes, amplifies noise), integration = low-pass (smooths).

Where it all goes: finding where a derivative = 0 → optimization & gradient descent (training every AI and fitting every neuron model). Adding up tiny changes → simulating neurons over time (the next quest: Differential Equations). One toolkit, the entire course. 🚀