14.5 The Gradient and directional derivatives For a function f(x,y), the rate of change in the x direction is given by fx, and the rate of change in the y direction is given by fy. These partial derivatives give rates of change in the directions of the vectors I and j, respectively to formally express a rate of change in any given direction, we will define the directional derivative Definition: The gradient of a function f(x,y) at a point p = (a,b) is the vector Introduction to the chain rule for paths We use the chain rule for paths when we are given a function F along a parametric path given by x(t) and y(t) in the plane or by x(t), y (t), and z(t) in 3-space A function F that is defined along a path r(t) results in a composition f(r(t)). The chain rule for paths is used to find the derivative of these composite functions rfp = hfx (a; b) ; fy (a; b)i In three variables, for f(x,y,z) and p = (a,b,c) Directional derivatives The derivative with respect to t of f(r(t)) at t=0 is called the directional derivative Of f with respect to u at P, and is denoted: rfp = hfx (a; b; c) ; fy (a; b; c) ; fz (a; b; c)i Duf (a; b) = d f (a + th; b + tk) ¡ f (a; b) f (r (t)) = lim t!0 dt t Duf (P) = rfp ¢ u Theorem 1: If f(x,y,z) and g(x,y,z) are differentiable and c is a constant, then i. ii. r (f + g) = rf + rg r (cf ) = crf iii. Product rule for gradients: r (f g) = f rg + grf iv. Chain rule for gradients: r (F (f (x; y; z))) = F 0 (f (x; y; z)) rf Theorem 2: If F and R(t) are differentiable, then d f (r (t)) = rfr(t) ¢ r0 (t) dt Properties of the gradient In other words, the rate of change in a given direction varies with the cosine of the angle theta between the gradient and the direction Duf (p) = rfp ¢ u = krfpk kuk cos µ = jjrfpk cos µ At each point on a level curve, the gradient vector must point in the direction in the domain of F that yields the steepest increase on the saddle. If we were actually on the saddle, each gradient vector tells us the horizontal direction we should take to go most steeply up the saddle