Section 10.5: The Chain Rule, Directional Derivatives, and the Gradient Vector
Theorem: (The Chain Rule for One Parameter)
Suppose that z = f (x, y) is a differentiable function of x and y, where x = x(t) and y = y(t)
are differentiable functions of a parameter t. Then z is a differentiable function of t and
∂z dx ∂z dy
dz
=
+
.
dt
∂x dt
∂y dt
Example: Let z = ex sin y, where x(t) = t and y(t) = t3 . Find
Example: Let z = xey , where x(t) = et and y = t2 . Find
1
dz
when t = 1.
dt
dz
when t = 0.
dt
Directional Derivatives
Recall that if f is a differentiable function of x and y and z = f (x, y), then the partial
derivatives fx (x, y) and fy (x, y) give the rates of change of z in the directions of x and y,
respectively.
Suppose we want to find the rate of change of z in the direction of an arbitrary unit vector
~u = hu1 , u2 i. Consider the surface S defined by z = f (x, y) and let z0 = f (x0 , y0 ) so that
P = (x0 , y0 , z0 ) lies on S. The vertical plane passing through (x0 , y0 , z0 ) in the direction of
~u intersects S in a curve C. The slope of the tangent line to C at P is the rate of change of
z in the direction of ~u.
Definition: The directional derivative of f at (x0 , y0 ) in the direction of a unit vector
~u = hu1 , u2 i is given by
f (x0 + u1 h, y0 + u2 h) − f (x0 , y0 )
,
h→0
h
D~u f (x0 , y0 ) = lim
provided that this limit exists.
Theorem: (Directional Derivative Formula)
If f is a differentiable function of x and y, then f has a directional derivative for any unit
vector ~u = hu1 , u2 i and
D~u f (x, y) = fx (x, y)u1 + fy (x, y)u2 .
Note: The vector that specifies the direction must be a unit vector.
2
Example: Find the directional derivative of f (x, y) = x2 sin y at the point (−1, 0) in the
direction of the vector ~v = h2, −1i.
Example: Find the directional derivative of f (x, y) = 4xy + y 2 at the point P = (−1, 1) in
the direction of the point Q = (3, 2).
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The Gradient Vector
Definition: If f (x, y) is a differentiable function, then the gradient of f is the vector function
∇f (x, y) = hfx (x, y), fy (x, y)i.
Example: Find the gradient of f (x, y) = cos(3x2 − 2y 2 ).
Example: Find the gradient of f (x, y) = x(x2 − y 2 )2/3 .
Example: Find the gradient of f (x, y) = exy + sin(x2 + 2y).
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Theorem: (Gradient Formula for the Directional Derivative)
If f is a differentiable function of x and y, then
D~u f (x, y) = ∇f (x, y) · ~u.
Example: Find the directional derivative of f (x, y) = x3 y 2 at the point (2, 3) in the direction
of the vector ~v = h−2, 1i.
Example: Find the directional derivative of f (x, y) = ex−y at the point P = (2, 2) in the
direction of the point Q = (1, −1).
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Theorem: (Properties of the Gradient)
Suppose that f (x, y) is a differentiable function.
1. At each point (x0 , y0 ), the function f (x, y) increases most rapidly in the direction of the
gradient vector ∇f (x0 , y0 ). Moreover, the maximum value of the directional derivative
D~u f (x0 , y0 ) is ||∇f (x0 , y0 )||.
2. The gradient vector ∇f (x0 , y0 ) is orthogonal to the level curve through (x0 , y0 ).
Example: In what direction does f (x, y) = ln(x2 + y 2 ) increase most rapidly at (1, 1)? What
is the maximum rate of change at (1, 1)?
Example: Find a unit vector that is orthogonal to the level curve of the function
f (x, y) = x2 +
at the point (1, 3).
6
y2
9