Unraveling the Mystery of Directional Derivatives: A Step-by-Step Guide

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In summary, the conversation is about a question that has stumped the speaker regarding finding the gradient of a function at a specific point. The tangent vector to a level curve is given, as well as the directional derivative in a specific direction. The speaker is unsure of how to find the gradient and asks for help in evaluating the problem. They mention that the answer provided by others is (1,-2), but they are unable to reach the same result themselves. They request assistance and ask for a step-by-step explanation.
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cooev769
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I've come across a question which as really stumped me because I thought I knew how to do this but apparently not.

The question is that we have a tangent vector to a level curve of a function of two variables f(x,y) at a point P is (2,1). Hence by my logic this means grad of f x unit vector of (2,1) is 0. The next part is that the directional derivative of f at P in the direction (3/5, 2/5) is 1/5. Hence by applying the same operation above. Find the gradient of f at P. I would just evaluate both and use simultaneous equations. But the answer they arrive at is really nice at (1, -2) I can't get it myself. Any help would be appreciated thanks.
 
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Looks like the method I'd use. Please show your working - step by step.
Double check the algebra and arithmetic.
 

1. What are directional derivatives?

Directional derivatives are a type of derivative in multivariable calculus that measures the rate of change of a function in a specific direction. It tells us how much a function changes when moving in a particular direction from a given point.

2. How are directional derivatives calculated?

To calculate a directional derivative, we use the gradient vector of the function and a direction vector to find the dot product. The result of the dot product is the directional derivative in that direction.

3. What is the significance of directional derivatives?

Directional derivatives are important in optimization problems where we want to find the maximum or minimum value of a function. They also have applications in physics, engineering, and other fields where understanding the rate of change of a function is crucial.

4. How do directional derivatives relate to partial derivatives?

Directional derivatives can be thought of as a generalization of partial derivatives. While partial derivatives measure the rate of change of a function in the x and y directions, directional derivatives measure the rate of change in a specific direction.

5. What is the difference between a positive and negative directional derivative?

A positive directional derivative indicates that the function is increasing in the given direction, while a negative directional derivative indicates that the function is decreasing. A zero directional derivative means that the function is not changing in that direction.

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