![]() Suppose we have a magical oven, with coordinates written on it and a special display screen: I’m a big fan of examples to help solidify an explanation. Likewise, with 3 variables, the gradient can specify and direction in 3D space to move to increase our function. If we have two variables, then our 2-component gradient can specify any direction on a plane. However, now that we have multiple directions to consider ($x$, $y$ and $z$), the direction of greatest increase is no longer simply “forward” or “backward” along the $x$-axis, like it is with functions of a single variable. The gradient of a multi-variable function has a component for each direction.Īnd just like the regular derivative, the gradient points in the direction of greatest increase ( here's why: we trade motion in each direction enough to maximize the payoff). The regular, plain-old derivative gives us the rate of change of a single variable, usually $x$. ![]() Now that we know the gradient is the derivative of a multi-variable function, let’s derive some properties. “Gradient” can refer to gradual changes of color, but we’ll stick to the math definition if that’s ok with you. Yes, you can say a line has a gradient (its slope), but using "gradient" for single-variable functions is unnecessarily confusing. ![]() The term "gradient" is typically used for functions with several inputs and a single output (a scalar field). Is zero at a local maximum or local minimum (because there is no single direction of increase).Points in the direction of greatest increase of a function ( intuition on why).The gradient is a fancy word for derivative, or the rate of change of a function.
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