the gradient of a function is in other words, .
many people think of the gradient as the derivative. this is unwise, encouraging confusion of row matrices and column vectors. the transpose is not as innocent as it looks; geometrically, it is only defined if we are in a space with an inner product. when there is no natural inner product, the vector is not natural, unlike the derivative, which as a linear transformation depends on no such choices.
the gradient of a function is a vector that points in the direction in which increases fastest.