14.6 Multivariable calculus chain rules In the cases of two and three variables, this chain rule states: d @f @f @f dx @f dy f (r (t)) = h ; i ¢ hx0 (t) ; y 0 (t)i = + dt @x @y @x dt @y dt d @f @f @f @f dx @f dy @f dz f (r (t)) = h ; ; i ¢ hx1 (t) ; y 0 (t) ; Z 0 (t)i = + + dt @x @y @z @x dt @y dt @z dt Next, lets consider the case of more general composite functions. Suppose, for example, that x,y,z are differentiable functions Of s and t - say, x = x(s,t), y = y(s,t), and z = z(s,t). The composition is then a function of S and t. We refer to S and t as The Independent variables Implicit differentiation In single-variable calculus, we used implicit differentiation to compute dy/dx when y is defined simplicity as a function of X through an equation f(x,y) = o we may not be able to solve explicitly for z(x,y), but we can treat f(x,y,z) as a composite function with X and y as independent variables and use the chain rule to differentiate implicitly with respect to x: @F @x @F @y @F @z + + =0 @x @x @y @x @z @x