The Directional Derivative

The Directional Derivative

Suppose z=f(x,y) is a differentiable at the point(a,b,c).

The Directional Derivative

Suppose z=f(x,y) is a differentiable at the point(a,b,c).

Let u be a unit vector pointing in the directionin which we want to find the derivative.

ˆ ˆu xi y j= Δ + Δ

( )udirectional derivative D f=

xΔ yΔ ˆ ˆu xi y j= Δ + Δ

1zΔ

2zΔ

1 2( )uD f z z= Δ + Δ

( , , )a b c

The Directional Derivative

xΔ yΔ ˆ ˆu xi y j= Δ + Δ

1zΔ

2zΔ

1 2( )uz zD f z z x yx y∂ ∂

= Δ + Δ = Δ + Δ∂ ∂

( , , )a b c

The Directional Derivative

1 2( )uD f z z= Δ + Δ

xΔ yΔ ˆ ˆu xi y j= Δ + Δ

1zΔ

2zΔ

1 2( )uz zD f z z x y f ux y∂ ∂

= Δ + Δ = Δ + Δ = ∇ ⋅∂ ∂

( , , )a b c

The Directional Derivative

1 2( )uD f z z= Δ + Δ

(A consequence is t ) chat os cos . uD f f u f u fθ θ= ∇ ⋅ = ∇ = ∇

(A consequence is t ) chat os cos . uD f f u f u fθ θ= ∇ ⋅ = ∇ = ∇

In what direction defined by i ( ) cos s at its maximum?uD f f θθ = ∇

(A consequence is t ) chat os cos . uD f f u f u fθ θ= ∇ ⋅ = ∇ = ∇

In what direction defined by i ( ) cos s at its maximum?uD f f θθ = ∇

0 0 radiansθ = ° =

(A consequence is t ) chat os cos . uD f f u f u fθ θ= ∇ ⋅ = ∇ = ∇

In what direction defined by i ( ) cos s at its maximum?uD f f θθ = ∇

0 0 radiansθ = ° =

In what direction defined by i ( ) cos s at its minimum?uD f f θθ = ∇

(A consequence is t ) chat os cos . uD f f u f u fθ θ= ∇ ⋅ = ∇ = ∇

In what direction defined by i ( ) cos s at its maximum?uD f f θθ = ∇

0 0 radiansθ = ° =

In what direction defined by i ( ) cos s at its minimum?uD f f θθ = ∇

180 radiansθ π= ° =

(A consequence is t ) chat os cos . uD f f u f u fθ θ= ∇ ⋅ = ∇ = ∇

In what direction defined by i ( ) cos s at its maximum?uD f f θθ = ∇

0 0 radiansθ = ° =

In what direction defined by i ( ) cos s at its minimum?uD f f θθ = ∇

180 radiansθ π= ° =

In what direction defined by is ( ) cos 0?uD f fθ θ= ∇ =

(A consequence is t ) chat os cos . uD f f u f u fθ θ= ∇ ⋅ = ∇ = ∇

In what direction defined by i ( ) cos s at its maximum?uD f f θθ = ∇

0 0 radiansθ = ° =

In what direction defined by i ( ) cos s at its minimum?uD f f θθ = ∇

180 radiansθ π= ° =

In what direction defined by is ( ) cos 0?uD f fθ θ= ∇ =

902

radiansπθ = ° =

Now apply what we've done to the picture below.

Now apply what we've done to the picture below.

The gradient tells you what direction to move in in the -plane in order to make your output increase as quickly as possible. This direction maximizes the directionalderivative.

xy

Now apply what we've done to the picture below.

The maximum rate of change at a point is equal to .f∇

Now apply what we've done to the picture below.

The maximum rate of change at a point is equal to .f∇

And the minimum rate of change is equal to - .f∇

All of this applies, also, to functions of the form ( , , ).w f x y z=

All of this applies, also, to functions of the form ( , , ).w f x y z=

If you move in the direction of the gradient vector in 3-dimensionalspace, then your values for ( , , ) will increase at their mostrapid rate.

w f x y z=

All of this applies, also, to functions of the form ( , , ).w f x y z=

And the maximum rate of change is equal to .f∇

All of this applies, also, to functions of the form ( , , ).w f x y z=

And the minimum rate of change is equal to - .f∇