Why does this work? Directional Derivatives.

In summary, the directional derivative of f(x,y) at the point (x,y,z) can always be solved using the equation Du▽f(x,y,z) = <fx(x,y), fy(x,y), <fx(x,y), fy(x,y)><fx(x,y), fy(x,y)>>, which is a variant of the chain rule. This rule multiplies the partial derivatives by the directional vector and is valid for any function f(x,y,z).
  • #1
DavidAp
44
0
Find the directional derivative of:
f(x,y) = (x^2)y-y^3, (2,1,3)


Can somebody explain to me why we did the step we did?

First we found the directional derivative of f(x,y,z)
▽f(x,y)
= <2xy, (x^2)-3(y^2), a>
▽f(2,1)
= <2(2)(1), (2^2)-3(1^2)), a>
= <2(2), 4-3, a> = <4,1, a>

and then we... did some sorcery mathematics?!
f(x,y) = (x^2)y-y^3
z = f(x, y)
F(x,y,z) = f(x,y) - z
▽F(x,y,z) = <2xy, (x^2)-3(y^2), -1>
▽F(2,1,3) = <4,1,-1>

A dot product for some reason...
<4,1,a><4,1,-1> = 0
4(4)+1(1)-a = 0
16+1 = a
17 = a

Therefore, the directional derivative is, <4,1,17>.
__________________________________________

My question is: if we are given a function f(x,y) and are told to find the directional derivative at the point (xo,yo,zo) is it safe to say that we can always solve it like we did above?

Du▽f(x,y,z) = <fx(x,y), fy(x,y), <fx(x,y), fy(x,y)><fx(x,y), fy(x,y)>>?

Can somebody explain to me why the equation above is valid? This isn't homework, I'm just reviewing my notes and couldn't understand what happened here. This is just for my own understanding... and because of that I'm worried I'm posting this in the wrong thread.

Thank you for taking the time to read my question.
 
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  • #2
This is just a variant of the chain rule. The partials are the directional derivatives in the direction of the standard x,y,z axes, so that the chain rule just multiplies by 1 at most.

Basically, when you do the quotient [f(x+th,y,z)-f(x,y,z)]/h as h->0, the chain rule
will multiply the partial by the directional vector.
 
  • #3
Bacle2 said:
This is just a variant of the chain rule. The partials are the directional derivatives in the direction of the standard x,y,z axes, so that the chain rule just multiplies by 1 at most.

Basically, when you do the quotient [f(x+th,y,z)-f(x,y,z)]/h as h->0, the chain rule
will multiply the partial by the directional vector.
So that equation, Du▽f(x,y,z) = <fx(x,y), fy(x,y), <fx(x,y), fy(x,y)><fx(x,y), fy(x,y)>>, will always hold (when asked for the directional derivative of f(x,y) at point (x,y,z))?
 

1. Why is the direction of a directional derivative important?

The direction of a directional derivative is important because it tells us how a function is changing in a specific direction. It allows us to understand the rate of change of a function in a particular direction, which is useful for optimizing functions and solving problems in fields such as physics, engineering, and economics.

2. What is the difference between a directional derivative and a partial derivative?

A directional derivative is the rate of change of a function in a specific direction, while a partial derivative is the rate of change of a function with respect to one of its variables. In other words, a directional derivative considers the change in a function along a specific direction, while a partial derivative considers the change in a function along one of its axes.

3. How do you calculate a directional derivative?

The directional derivative can be calculated using the gradient of the function and the unit vector in the desired direction. It is given by the dot product of the gradient and the unit vector. Mathematically, it can be expressed as: Duf(x,y) = ∇f(x,y) ⋅ u, where u is the unit vector in the desired direction.

4. What is the significance of the directional derivative in real-world applications?

The directional derivative has many real-world applications, such as determining the optimal path for a plane or missile, finding the steepest descent/ascent in a terrain, and analyzing the movement of fluids in a specific direction. It is also used in economics to optimize production processes and in physics to study the behavior of particles in a specific direction.

5. Can the directional derivative be negative?

Yes, the directional derivative can be negative. A negative directional derivative indicates that the function is decreasing in the direction of the unit vector. This can be useful in optimization problems, where we want to minimize a function. A negative directional derivative can also represent a downhill slope in a terrain or a decrease in temperature in a specific direction.

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