gradient

the gradient of a function $\text{[math]}$ is
$\text{[math]}$ in other words, $\text{[math]}$ .

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 $\text{[math]}$ is not natural, unlike the derivative, which as a linear transformation depends on no such choices.
[cite:@calc_hubbard_2015 chapter 1 vectors, matrices, and derivatives]

the gradient of a function $\text{[math]}$ is a vector that points in the direction in which $\text{[math]}$ increases fastest.