partial derivative
/ PAR-shul duh-RIV-uh-tiv /
Most interesting things depend on several inputs at once. A cake's tastiness depends on sugar, baking time, and oven temperature together. A partial derivative isolates just one of those knobs: it asks 'if I change only the sugar, holding baking time and temperature perfectly still, how fast does tastiness change?' You freeze everything but one variable, then measure the ordinary slope along that single direction.
It is written with a curly symbol (∂) to flag that other variables are lurking but held fixed. A function of many inputs therefore has many partial derivatives — one per input. Collect them all into a single list and you get the gradient, the vector that points in the direction of steepest increase. So the gradient is just all the partial derivatives stacked side by side.
This is the bread and butter of training neural networks. A modern model has millions or billions of internal numbers, and learning means asking, for each one separately, 'how does the total error change if I nudge just this number?' Each answer is a partial derivative. Backpropagation is a clever bookkeeping scheme for computing all of them efficiently in one sweep. The honest caveat: a partial derivative deliberately ignores how the variables might move together in the real world; it answers a strict 'all else held equal' question, which is powerful precisely because it is so narrow.
For f(x, y) = x² + 3y, the partial derivative with respect to x is 2x (treating y as a frozen constant, so 3y vanishes), and with respect to y it is just 3 (treating x as frozen). Stack them and the gradient is (2x, 3): at the point (1, 1) it points in the direction (2, 3), the steepest way uphill.
One knob at a time: freeze every variable but one, measure its slope, then stack the answers into the gradient.
The gradient (the bundle of all partial derivatives) points toward steepest ascent; gradient descent simply walks the opposite way, downhill. Remember that each partial derivative holds every other variable rigidly fixed — a real, useful fiction.