Math 2204 Multivariable Calculus – Chapter 14: Partial Derivatives Sec. 14.6: Directional Derivatives and Gradient Vectors

I . Directional Derivatives

A. What if you want to calculate the slope at any point moving in any direction, not just in the direction of x or y.

1. Find the slope moving in NE direction from (4,-6)

-8 -6 -4 2 80 140 209 4 71 124 186 6 58 102 152

2. In general, slope from (x1, y1) to (x2 , y2 ) is m =f (x2 , y2 ) − f (x1, y1)(x2 − x1)

2 + (y2 − y1)2

B. Definition The derivative of f at P0 (x0 , y0 ) in the direction of the unit vector

u = u1i + u2

j is the

number D!u f (P0 ) = limh→0f (x0 + hu1, y0 + hu2 )− f (x0, y0 )

h⎛⎝⎜

⎞⎠⎟ , provided the limit exists.

C. Theorem If f is a differentiable function of x and y, then f has a directional derivative in the direction of any unit vector

u = u1i + u2

j and D

u f (x, y) = fx (x, y)u1 + fy (x, y)u2

D. Example: Find the directional derivative Du f (x, y) if f (x, y) = x

3 − 3xy + 4y2 in the

direction of v = 3

i +j at the point (1,2).

E↓

N →

II. Gradient Vectors A. Definition

The gradient vector (gradient) of f(x,y) at a point P0 (x0 , y0 ) is the vector ∇f =

∂f∂xi +

∂f∂yj

obtained by evaluating the partial derivatives of f at P0 . B. The notation ∇f is read as “del f” or “grad f” or “gradient of f” C. Theorem: The Directional Derivative is a Dot Product

If f(x,y) is differentiable in an open region containing P0 (x0 , y0 ) , then D!u f (x0, y0 ) = ∇f (P0 ) ⋅

!u , the dot product of the gradient f at P0 and u .

D. Algebra Rules for Gradients 1. Constant Multiple Rule: ∇(kf ) = k∇f , (any number k)

2. Sum/Difference Rule: ∇( f ± g) = ∇f ±∇g

3. Product Rule: ∇( fg) = f∇g ± g∇f

4. Quotient Rule: ∇fg

⎛⎝⎜

⎞⎠⎟=g∇f − f∇g

g2

E. The gradient vector points in the direction in which f increases most rapidly. F. The vector’s magnitude is the rate of change in that direction. G. If the directional derivative of f at (a,b) is zero in every direction, ∇f = 0 . H. At every point (x0 , y0 ) in the domain of a differentiable function f(x,y), the gradient of f is

normal to the level curve through (x0 , y0 ) .

I. The gradient vector is normal to the tangent line. J. The gradient doesn’t necessarily point to the steepest point (peak); it goes to the next

contour in the shortest distance.

K. ∇f is large when contours are close together and small when they are far apart.

L. Examples

1. Find the derivative of F(x, y) = 3x2 − xy + y2 in the direction of u = 1

2

i + 1

2

j at (1,5).

2. Given F(x, y) = 2 + 1

2 x − y and the point P0 (2,0) .

a. Find the derivative in the direction of v =i + 2

j at P0 .

b. Find the equation of the tangent line at P0 . 3. Find the derivative of F(x, y, z) = xy + ln z in the direction of

v =i + 4

k at (2,-1,1).

III. Properties of the Directional Derivative Du f = (∇f ) ⋅u = ∇f cosθ

A. Properties 1. The function f increases most rapidly when cosθ = 1 or θ = 0 or when

u is the direction of ∇f . That is, at each point P in its domain, f increases most rapidly in the direction of the gradient vector ∇f at P. The derivative in this direction is

Du f = ∇f cos0 = ∇f .

2. The function f decreases most rapidly when cosθ = −1 or θ = π or when u is the

direction of −∇f . The derivative in this direction is Du f = ∇f cosπ = − ∇f .

3. Any direction u orthogonal to a gradient ∇f ≠ 0 is a direction of zero change in f

because cosθ = 0 or θ =π2

and Du f = ∇f cosπ

2= ∇f ⋅0 = 0

B. Example 1. Suppose T (x, y) = 70 + xy represents level curves around a heat source. Let P(2,−1)

be a point where you are sitting. a. How hot is it? b. What is the equation of the level curve you are on? c. If you move in the direction of

v =< −1,1 > , is it getting hotter or cooler?

The direction in which f(x,y) increases most rapidly at (1,1) is the direction of

∇f (1,1) =

i +j . (the direction of

steepest ascent on the surface at (1,1,1))

d. Find the direction of maximum temperature increase/ decrease. e. What is the rate? f. What is the direction if you wish to stay at 680 ? g. Find the direction in which Du f = 1 .

IV. Tangent Planes and Normal Line A. Definitions 1. The tangent plane at the point P0 (x0 , y0 , z0 ) on the level surface f (x, y, z) = c of a

differentiable function f is the plane through P0 normal to ∇f P0 .

2. The normal line of the surface at P0 is the line through P0 parallel to ∇f P0 .

B. Equations 1. Tangent Plane to f (x, y, z) = c at P0 (x0 , y0 , z0 )

fx (P0 )(x − x0 ) + fy (P0 )(y − y0 ) + fz (P0 )(z − z0 ) = 0

OR Ax + By +Cz =

η ⋅P0 ⇒ fx (P0 ) x + fy (P0 ) y + fz (P0 ) z = ∇f (P0 ) ⋅

P0 , where

η = A,B,C = fx (P0 ) , fy (P0 ) , fz (P0 ) = ∇f (P0 )

2. Normal Line to f (x, y, z) = c at P0 (x0 , y0 , z0 )

x = x0 + fx (P0 )t y = y0 + fy (P0 )t z = z0 + fz (P0 )t

3. Tangent Plane to a Surface z = f (x, y) at (x0 , y0 , f (x0 , y0 ))

The tangent plane to the surface z = f (x, y) of a differentiable function f at the point

P0 (x0 , y0 , z0 ) = (x0 , y0 , f (x0 , y0 )) is fx (x0 , y0 )(x − x0 ) + fy (x0 , y0 )(y − y0 ) − (z − z0 ) = 0

C. Examples

1. Find the equation of the tangent plane and the normal line to the level surface x + y + z2 = 7 at the point P(1,2,−2) .

2. Find the equation of the tangent plane to the level surface xeyz2 + ln x = 4 at the point P(1,0,−2) .

3. Find the equation of the tangent plane to the curve z = x2 + y2 at the point P(3, 4) .