What a Partial Derivative Is

Stand on a hillside. 'How steep is it?' has no single answer — it depends on which way you face. Walk east and you might climb; walk north and you might descend. A surface z = f(x, y) has a different slope in every direction. The partial derivative answers the simplest version of the question: 'How steep is it if I walk in the x-direction only, keeping y fixed?'