<strong>Gradient</strong>, <strong>Chain</strong> <strong>Rule</strong>, <strong>and</strong> <strong>Directional</strong> <strong>Derivatives</strong> f ′ (x) y = f(x) f What’s going to replace the derivative of a function of one variable when is a real-valued function of two or more variables? Since slope now depends on direction, not just sign, we can expect vectors will be needed! In two variables Definition: the <strong>Gradient</strong> of a real-valued function whose components are the partial derivatives of z = f(x, y) ∂f ∂f (∇f)(x, y) = i + j ∂x ∂y f x y is the vector function in the – <strong>and</strong> -directions. This definition generalizes immediately to a function : y = f( x 1 , x 2 , … , x n ) n <strong>and</strong> to any function of variables: w = f(x, y, z) ∂f ∂f ∂f (∇f)(x, y, z) = i + j + k , ∂x ∂y ∂z ∂f ∂f ∂f (∇f)(x) = i 1 + i 2 + … + i n . ∂x 1 ∂x 2 ∂x n f( x 1 , x 2 , … , x n ) n n = 2, 3, … , (∇f)( x 1 , x 2 , … , x n ) So if is a function of variables, then the value of is a vector in R n . Later we shall use the name vector fields to describe functions having domain <strong>and</strong> range in R n . Example: for , To draw the graph of ∇f we select a set of points P(x, y) <strong>and</strong> represent by a vector with initial point <strong>and</strong> length scaled so that it’s not too long but remains a fixed proportion of the true length of (∇f)(P) for every . Drawing is not helpful, of course, because it would cover the plane with arrows, so we choose a representative set. But for functions of variables, the graph of f(x, y) = x 2 − y 2 (∇f)(x, y) = 2x i − 2y j . (∇f)(P) P ∇f (∇f)(P) or more is usually too cluttered to be of much use even if a computer graphics program is used. 3 P