Who is responsible for coming up with the gradient and why did they do so? In which work was it first described?
I have Googled this extensively, to no avail, and Boyer's History of Calculus does not appear to contain the answer to this.
The vector differential operator, now written $\nabla$ and called nabla or del, was introduced by William Rowan Hamilton (1805-1865). Hamilton wrote the operator as a [90°] rotated nabla and it was P.G.Tait who established $\nabla$ as the conventional symbol; see his An Elementary Treatise on Quaternions (1867). Tait was also responsible for establishing the term nabla.
See also :
The vector sum which is the resultant rate of increase of $V$ is denoted by $\nabla V$.
$\nabla V$ represents a directed rate of change of $V$ - a directed or vector derivative of $V$, so to speak. For this reason $\nabla V$ will be called the derivative of $V$; and $V$, the primitive of $\nabla V$. The terms gradient and slope of $V$ are also used for $\nabla V$.