Suppose we have a multivariate scalarvalued function f : 1R" i~ R. The directional derivative of f at x = [$1, . . . ,3\")? E R\" in the direction ole vector u = [uh . . . ,u)T E R", when it is dened. is the limit x + h\") - x} ( nix) H h = m ml walrus\" +huaml,...,ma hIv h lntuitivelv, this scalar quantity tells us the rate of change of the function f it we move in the direction of u from the point (1:1, . . . , :11\") in the domain. However, if f is known to be differentiable at I (in the sense dened in a previous question}, then the directional derivative of 3' exists along any vector u at x, and is simplyr given by the dot product (vumx) = [We] -u, where 3f 331 _> V f : : 3:\" is the gradient of f at x. Consider the function 3' : R2 > R dened by the rule ay] = 4:23? 8531', for z, y E JR. One can show that this function is differentiable everywhere. (it First, by taking partial derivatives, we note that the gradient of f at (21,39!) 2 (2, 2] is the column vector _. .. Wit2,4) = i In a. [ii] Therefore, taking the dot product (or othenvise}, we nd that the directional derivative V\" f at (2, 2] in the direction of the vector u : [1,2)T is the scalar area2) = l in a