12.6 Directional Derivatives and the Gradient Vector
Recall that the two partial derivatives fx and fy of a function z = f (x, y) represent
the rate of change of z in the x- and y - directions, that is, in the directions of the unit
vectors i and j. Now let's consider how to nd the rate of change of z in the direction
of an arbitrary unit vector u.
Denition. The directional derivative of f (x, y) at (x0, y0) in the direction of a
unit vector u = ha, bi is
f (x0 + ah, y0 + bh) − f (x0 , y0 )
h→0
h
Du (x0 , y0 ) = lim
if this limit exists.
Remark. If u = i = h1, 0i, then Dif = fx. If u = j = h0, 1i, then Djf = fy .
Theorem. If f is a dierentiable function of x and y, then f has a directional
derivative in the direction of any unit vector u = ha, bi and
Du f (x, y) = fx (x, y)a + fy (x, y)b
If the unit vector u makes an angle θ with the positive x-axis, then
Du f (x, y) = fx (x, y) cos θ + fy (x, y) sin θ
Example 1. Find the rate of change of f (x, y) = x3 + sin(xy) at the point (1, 0) in the
direction of the unit vector u given by angle θ = π/4.
Gradient
Denition. The gradient of f (x, y) is the vector function ∇f (or gradf ) dened by
∇f (x, y) = hfx (x, y), fy (x, y)i = fx (x, y)i + fy (x, y)j
Then
Du f (x, y) = ∇f (x, y) · u
for a unit vector u.
Example 2. Find the directional derivative of the function f (x, y) = xexy at the point
(−3, 0) in the direction of the vector v = 2i + 3j.
The Directional Derivative of Function of Three Variables
Denition. The directional derivative of f (x, y, z) at (x0, y0, z0) in the direction
of a unit vector u = ha, b, ci is
f (x0 + ah, y0 + bh, z0 + ch) − f (x0 , y0 , c0 )
h→0
h
Du (x0 , y0 , z0 ) = lim
if this limit exists.
The gradient vector of f (x, y, z) is
∇f (x, y, z) =hfx (x, y, z), fy (x, y, z), fz (x, y, z)i
=fx (x, y, z)i + fy (x, y, z)j + +fz (x, y, z)k
Theorem. If f is a dierentiable function of x,y and z , then f has a directional
derivative in the direction of any unit vector u = ha, b, ci and
Du f (x, y, z) = fx (x, y, z)a + fy (x, y, z)b + fz (x, y, z)c = ∇f (x, y, z) · u
Example 3. Find the directional derivative of the function f (x, y, z) = z 3 − x2 y at the
point (1, 6, 2) in the direction of the vector v = h1, −2, 3i.
Maximizing the Directional Derivative
Question: In which of all possible directions does f change fastest and what is the
maximum rate of change?
Theorem. Suppose f is a dierentiable function of two or three variables. The
maximum value of the directional derivative Du f is |∇f | and it occurs when u has
the same direction as the gradient vector ∇f .
Example 4. Suppose that the temperature at a point (x, y, z) in the space is given by
T (x, y, z) =
100
,
1 + x2 + y 2 + z 2
where T is measured in ◦ C and x, y, z in meters. In which direction does the temperature increase fastest at the point (1, 1, −1)? What is the maximum rate of increase?
Tangent Planes to Level Surfaces
For any constant k , suppose S is a surface with equation F (x, y, z) = k , that is, it is
a level surface of the function w = F (x, y, z)
Fact: The gradient vector ∇F (x0, y0, z0) is orthogonal to the level surface F (x, y, z) =
k at the point (x0 , y0 , z0 ).
The tangent plane to the surface F (x, y, z) = k at the point (x0 , y0 , z0 ) has the equation:
Fx (x0 , y0 , z0 )(x − x0 ) + Fy (x0 , y0 , z0 )(y − y0 ) + Fz (x0 , y0 , z0 )(z − z0 ) = 0
The normal line to the surface at the point (x0 , y0 , z0 ) is the line passing through
(x0 , y0 , z0 ) and perpendicular to the tangent plane. Its direction is given by the gradient
vector ∇F (x0 , y0 , z0 ) and symmetric equations are
x − x0
y − y0
z − z0
=
=
Fx (x0 , y0 , z0 ) Fy (x0 , y0 , z0 ) Fz (x0 , y0 , z0 )
Example 5. Find the equation of the tangent plane and normal line at the point
(1, 0, 5) to the surface xeyz = 1.