vela said:The problem wants you to find the directional derivative at all points. It's like asking you what g'(x) is as opposed to g'(1). Does that make sense?
A directional derivative is a type of derivative that measures the rate of change of a function in a specific direction. It is commonly used in multivariable calculus and can be thought of as the slope of a function in a certain direction.
The directional derivative is calculated by taking the dot product of the gradient vector of the function and a unit vector in the desired direction. This can also be written as the product of the magnitude of the gradient vector and the cosine of the angle between the gradient vector and the direction vector.
A directional derivative is significant because it allows us to understand how a function is changing in a specific direction. This can be useful in optimization problems, where we want to find the maximum or minimum value of a function in a given direction.
No, the directional derivative is not always defined. It depends on the differentiability of the function at a given point and the direction in which the derivative is being calculated. If the function is not differentiable or the direction is not well-defined, then the directional derivative may not exist.
Directional derivatives have many real-world applications, such as in physics, engineering, and economics. For example, in physics, directional derivatives can be used to calculate the rate of change of temperature or pressure in a specific direction. In economics, they can be used to understand the rate of change of production or demand in a certain direction.