Partial Derivatives

Multivariate calculus from first principles

One idea carries most of multivariable calculus: to differentiate a function of many variables, change only one variable at a time and freeze all the others. Hold y still, wiggle x, and ask how f responds. That rate of change is the partial derivative ∂f/∂x.

The curly ∂ ("partial") is the only new notation. Everything else is Course-I differentiation (power rule, product rule, chain rule) applied as if the frozen variables were just constants.

Stand on a hillside and the slope you feel depends on which way you face. Walk due east, holding your north-south position fixed, and the steepness underfoot is the partial derivative ∂f/∂x. Turn and walk due north instead, holding east-west fixed, and you feel a different slope, ∂f/∂y. Each partial freezes one direction and reports the rise or fall along the other.

Where this lives in MLPicture freezing every weight in a network except one, then asking how the loss moves as you nudge that single weight. The answer is the partial derivative ∂L/∂wᵢ: its sign tells you which way to push the weight to lower the loss, its size tells you how sensitive the loss is to it. Collect one partial per weight and you have the gradient, which the next few lessons assemble.

▶ Partial Derivatives

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