Conclusion

This chapter made change measurable.

You learned that:

• average slope compares change over an interval
• a derivative measures local sensitivity at one input
• partial derivatives isolate one input while holding others fixed
• gradients collect partial derivatives into a vector
• directional derivatives ask how fast a function changes along a chosen direction
• Jacobians track sensitivities for vector-output functions
• Hessians describe curvature, or how gradients change
• the chain rule multiplies local sensitivities through composed functions
• computation graphs show value dependencies
• backpropagation applies the chain rule backward through the graph

The Main Skill

The main skill is to read a training computation as a chain of local sensitivities.

When a model trains, every update comes from the same question:

How does the loss change when this quantity changes?

That question can be asked for one weight, one vector of weights, one intermediate activation, or one whole path through a computation graph.

The habit to keep is local and precise. A derivative is measured at a point. A partial derivative holds other inputs fixed. A gradient points uphill for the quantity being measured. Backpropagation carries these local facts through the graph without changing what they mean.

What Comes Next

Probability comes next.

Calculus explains change. Probability explains uncertainty. Together, they help us understand losses, distributions, likelihood, sampling, and model outputs.

Keep This Question Nearby

When a model trains, ask:

What is changing, with respect to what, and where is that sensitivity measured?