There are analogous rules for differentiation for multivariate functions as for univariate functions. For example, there is a multivariate version of the chain rule. Suppose f : IR -+ R" and g: R" + RP are differentiable functions. Notice that the codomain of f is the domain of g, so it makes sense to consider the composition of these functions, gof : R" + RP The multivariate chain rule states that the Jacobian matrix of go f at a point a R2 be the functions defined by the rules 6x2- 4y2 f(z, y, =) = Ary for I, y, # E R, and 5x + 2# 10u + 6wj2 g(u, u, w) = for u, U, WER, respectively. Additionally, suppose a = (5, 2,3) e IR3 (i) First, observe that the 3rd column of the Jacobian matrix of f at a, Ja f, is the array 1 = (ii) Next, we evaluate to find that, as a column vector, f(a) = Observe that the 2nd row of the Jacobian matrix of gat f(a), Jf(m)9. is the array V = (ill) Finally, by the multivariate chain rule, or otherwise, we find that the (2, 3) th entry of the Jacobian matrix of go f at a, Ja(go f), is W=