Then (a) the maximum directional derivative of f at (x0, y0) is |∇f(x0,y0)| and occurs for u with the same direction as ∇f(x0,y0), (b) the minimum directional derivative of f at (x0,y0) is −|∇f(x0,y0)| and occurs for u with the opposite direction as ∇f(x0, y0), and (c) the directional derivative of f at (x0,y0) is zero …

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## How do you find the minimum directional derivative?

Then (a) the maximum directional derivative of f at (x0, y0) is |∇f(x0,y0)| and occurs for u with the same direction as ∇f(x0,y0), (b) the minimum directional derivative of f at (x0,y0) is −|∇f(x0,y0)| and occurs for u with the opposite direction as ∇f(x0, y0), and (c) the directional derivative of f at (x0,y0) is zero …

## What is directional derivative example?

The directional derivative is maximal in the direction of (12,9). (A unit vector in that direction is u=(12,9)/√122+92=(4/5,3/5).) (b) The magnitude of the gradient is this maximal directional derivative, which is ∥(12,9)∥=√122+92=15. Hence the directional derivative at the point (3,2) in the direction of (12,9) is 15.

**How do you find the minimum rate of change of a directional derivative?**

To get minimum rate of increase per unit distance you should move in the direction opposite ∇f(a, b). Then the rate of increase per unit distance is −|∇f(a, b)|. ◦ The directions giving zero rate of increase are those perpendicular to ∇f(a, b).

### What does the directional derivative tell you?

Directional derivatives tell you how a multivariable function changes as you move along some vector in its input space.

### What are directional derivatives used for?

**How do you interpret a directional derivative?**

The concept of the directional derivative is simple; Duf(a) is the slope of f(x,y) when standing at the point a and facing the direction given by u. If x and y were given in meters, then Duf(a) would be the change in height per meter as you moved in the direction given by u when you are at the point a.

#### In which direction is the directional derivative the largest?

The maximum value of the directional derivative occurs when ∇ f ∇ f and the unit vector point in the same direction.

#### Why do we need directional derivatives?

The directional derivative allows us to find the instantaneous rate of z change in any direction at a point. We can use these instantaneous rates of change to define lines and planes that are tangent to a surface at a point, which is the topic of the next section.

**What is directional derivative?**

The rate of change of f(x, y) in the direction of the unit vector →u = ⟨a, b⟩ is called the directional derivative and is denoted by D→uf(x, y). The definition of the directional derivative is,

## How do you find the directional derivative of a unit vector?

For instance, the directional derivative of f (x,y,z) f ( x, y, z) in the direction of the unit vector →u =⟨a,b,c⟩ u → = ⟨ a, b, c ⟩ is given by, Let’s work a couple of examples.

## How do you find the derivative of a function with two variables?

There are similar formulas that can be derived by the same type of argument for functions with more than two variables. For instance, the directional derivative of f (x,y,z) f (x, y, z) in the direction of the unit vector →u =⟨a,b,c⟩ u → = ⟨ a, b, c ⟩ is given by,

**What is the derivative of u1 t + x 0?**

Here we have used the chain rule and the derivatives d d t ( u 1 t + x 0) = u 1 and d d t ( u 2 t + y 0) = u 2 . The vector ⟨ f x, f y ⟩ is very useful, so it has its own symbol, ∇ f, pronounced “del f”; it is also called the gradient of f .