Question

Find the directional derivative at a point (which means find the direction unit vector) REVISE?????

a) The directional derivative depends solely on the magnitude of the gradient vector
b) The directional derivative is always a negative value
c) The directional derivative is independent of the direction unit vector
d) The directional derivative can be found using the dot product of the gradient and direction vectors

Answer 1

The directional derivative at a point is found by using the dot product of the gradient vector and a direction unit vector reflecting the rate of change of the function in a specified direction.

Step-by-step explanation:

The directional derivative at a point can be computed by taking the dot product of the gradient vector of a function at that point and a direction unit vector. This derivative represents the rate at which the function is changing at that point in the specified direction. The magnitude of the gradient vector influences the directional derivative's value but it also crucially depends on the direction in which you're taking the derivative; the steepest ascent is in the direction of the gradient vector itself.
Therefore the correct statement from the options provided is (d) "The directional derivative can be found using the dot product of the gradient and direction vectors". It is important to note that the directional derivative is not necessarily always negative and it does vary based on the chosen direction unit vector.