Mathematics
Partial Derivatives and Gradients for Functions of Several Variables
Quick fact
The gradient vector points in the direction of the steepest increase of a function, and its magnitude is exactly the rate of that steepest ascent—this fact powers everything from hill-climbing algorithms to the backpropagation in neural networks.
Why this is interesting
You know that the slope of a hill tells you how steep it is in one direction, but what if you're standing on a mountain and need to know the steepest way up? How does calculus tell you which direction to walk?